Problem 52973. Easy Sequences 45: Second Derivative of Inverse Polynomial Function

Created by Ramon Villamangca

Solve Later

{"parts":[{"partUri":"/matlab/document.xml","contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?><w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>The </w:t></w:r><w:hyperlink w:docLocation=\"https://en.wikipedia.org/wiki/Inverse_function\"><w:r><w:t>inverse of a function</w:t></w:r></w:hyperlink><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>f(x)</w:t></w:r></w:customXml><w:r><w:t>, is the function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>f^{-1}(x)</w:t></w:r></w:customXml><w:r><w:t>, that reverses </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>f</w:t></w:r></w:customXml><w:r><w:t>. That means that if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>f(a)=b</w:t></w:r></w:customXml><w:r><w:t>, then </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>f^{-1}(b)=a</w:t></w:r></w:customXml><w:r><w:t>. For example, the function to convert celsius temperature to fahrenheit is: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>T(x) = \\frac_{9}_{5}x+32</w:t></w:r></w:customXml><w:r><w:t>, the inverse function (convert from fahrenheit to celsius) is: </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>T^{-1}(x) = \\frac_{5}_{9}(x-32)</w:t></w:r></w:customXml><w:r><w:t>. So that, </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>T(100)=212</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>T^{-1}(212)=100</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>Given a polynomial function </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>P</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t> (presented as vector of numbers), and a value </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Q(x)=P^{-1}(x)</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, write a program that evaluates </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R(n)</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>, where </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R(x)=\\frac{\\mathrm{d^2Q} }{\\mathrm{d} x^2}</w:t></w:r></w:customXml><w:r><w:rPr><w:b/></w:rPr><w:t>. </w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:t>For example, if </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>P = [3\\ -7\\ 2]=3x^2-7x+2</w:t></w:r></w:customXml><w:r><w:t>, and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>n=5</w:t></w:r></w:customXml><w:r><w:t>, then </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>Q(x)=P^{-1}(x) = \\frac{7}{6} \\pm \\frac{\\sqrt{12 x + 25}}{6}</w:t></w:r></w:customXml><w:r><w:t> and </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R(x)=\\frac{\\mathrm{d^2Q} }{\\mathrm{d} x^2}=\\pm\\frac{6}{\\left(12x+25\\right)^{3/2}</w:t></w:r></w:customXml><w:r><w:t>. Therefore </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R(n) = R(5) \\approx [-0.0077\\ 0.0077]</w:t></w:r></w:customXml><w:r><w:t>.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/><w:jc w:val=\"left\"/></w:pPr><w:r><w:rPr><w:b/></w:rPr><w:t>NOTE: </w:t></w:r><w:r><w:t>It is possible for </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R(n)</w:t></w:r></w:customXml><w:r><w:t> to return some complex numbers. We are interested only with real values, so in cases where there are no real </w:t></w:r><w:customXml w:element=\"equation\"><w:customXmlPr><w:attr w:name=\"displayStyle\" w:val=\"false\"/></w:customXmlPr><w:r><w:t>R(n)</w:t></w:r></w:customXml><w:r><w:t>, please output an empty vector. Also please round-off your output to 4 decimal places, and sorted in ascending order.</w:t></w:r></w:p></w:body></w:document>","relationship":null}],"relationships":[{"relationshipType":"http://schemas.mathworks.com/matlab/code/2013/relationships/document","target":"/matlab/document.xml","relationshipId":"rId1"}]}

<div style = "text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; "><div style="block-size: 299.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 149.75px; transform-origin: 407px 149.75px; vertical-align: baseline; "><div style="block-size: 77px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 38.5px; text-align: left; transform-origin: 384px 38.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.5px 8px; transform-origin: 14.5px 8px; unicode-bidi: normal; ">The <a target='_blank' href = "https://en.wikipedia.org/wiki/Inverse_function">inverse of a function</a><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEEAAAAlCAYAAAAdkoQJAAACWklEQVRoge2ZXbWsMAyFPw84wAAGUDAKxsE4wMFYQAMS8IAFNGBh7kObRQ6HKbRNOfeBvVYfBqZtmp+dNMCNGzdu/F94AZXRWi3QGK11GXrgabheBQzGa57euAcmPwacRY7Q47yghDwTFyqiBmY/Kr/xR/3+hg4YC8pVeRkuCY0Jd2jZ7OV/L4E5rf9PXVY0Xjj5ikKsvrVoS9gLRlwoXIEZ53VFN/gQF9dXeYGgI+yVWZDD6FA4g4ELXFShxslYhCR7jmN/i8rPeZcQKIAZp3wTNDgPaFlDYVLPjrjg4ec8EvauA+sfFUgDhiEx+iEZQVKhPD+ycOfnnKkhwB2sZ1X41osa9W7hO8/IvqY8JGkw1qqDn3MWNavl5bBSf9S4g7/9s5ClY5V/CsIHH+Jq/pE4JWjIQcSiMRWhkLgpOWo+iEGOEnQ2mogjV5lrVi8Iw6ewfI4SUPvGkpy5ErRFYt0rVwkyP9YDc7LSLraxGQPhklQIsS7EcZE5MYo15oS5OcLojBRrVdnXqnHD4hdMqcBSWbph9bwUPhpJM9outBCpzRDJ7WchfQGx/B4v9IRDc8Hw1ipX59hLk4ZUgCG8/Ghwh9WsvuWknrBSxfvM+OBNWorSOBJKZ5+9sGsO3m9xRulREFfMvZGFmirSrxz5nte7g/d6rQXD1Air9nNdS+p+01p+B2+M+5hScFhdSZ+4eDdLWxu0xNcTv/Dkp6Wk0LG8hHSU6TWadJp1x7hija0SbbEntoqocCGQ3WoX159x8SvFRknXtfwMZybnA5cF/uST1o0bN278Jf4BiS/WMxlysVkAAAAASUVORK5CYII=" style="width: 32.5px; height: 18.5px;" width="32.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 50px 8px; transform-origin: 50px 8px; unicode-bidi: normal; ">, is the function <img src="data:image/png;base64,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" style="width: 44px; height: 19.5px;" width="44" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 47.5px 8px; transform-origin: 47.5px 8px; unicode-bidi: normal; ">, that reverses f<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 63px 8px; transform-origin: 63px 8px; unicode-bidi: normal; ">. That means that if <img src="data:image/png;base64,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" style="width: 58.5px; height: 18.5px;" width="58.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20px 8px; transform-origin: 20px 8px; unicode-bidi: normal; ">, then <img src="data:image/png;base64,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" style="width: 70.5px; height: 19.5px;" width="70.5" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.8667px 8px; transform-origin: 15.8667px 8px; unicode-bidi: normal; ">. For example, the function to convert celsius temperature to fahrenheit is: <img src="data:image/png;base64,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" style="width: 97.5px; height: 28px;" width="97.5" height="28"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 112px 8px; transform-origin: 112px 8px; unicode-bidi: normal; ">, the inverse function (convert from fahrenheit to celsius) is: <img src="data:image/png;base64,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" style="width: 121px; height: 28px;" width="121" height="28"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 30.5px 8px; transform-origin: 30.5px 8px; unicode-bidi: normal; ">. So that, <img src="data:image/png;base64,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" style="width: 88px; height: 18.5px;" width="88" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMgAAAAnCAYAAABKSgfJAAAEn0lEQVR4nO2c0ZXjIAxFXw/uwA2kganAFaSDdOAOtoXU4BLcw7SQGtLC7AfWWnHASIhJcFb3HP/MGAuEBEJAAMdxHMdxHMdxHMdxHCEnADOAP++uiOO0RA9gBHAH8IMyB7kA6GpW6g30AIZ3V8Jpjx7BuEeUOcgVwLl2pd7EiNAepyG+jE9fqR4lDnJFmD0kUH1PynpxyJk1aOVe8ZlOUqK7Hqv+SuSVlv3HCcEo6bkjrANmALfN/1J/f5eDjEt99ugQjO0Hz+2UOhYQ2kjfkRi6Ve6Mz5kVtboDQttvACassyrpLudklrJPXJaKT3g29Blrx26N9rz8/VsrcAeNg3wh75w91nVN6snJihl6rpNryD0t36g1+LyDEt2BldnqiGx1RtrQLWWjTIiPwh0eGxZbPH5HKmJB4yAz8mEIzXZ8JD7hudNSC+NhqdOA0FZpJ1vlElfkZ8hWKdUdGfINcUOmQTtmI5aySX4SHxvw2JkxZtTNukgdRDJ7DAgjcKpDSBbNntK65Tq5plxqp2XN1AJS3XXIZzIpctl+y1I2SY90nPuHfSw1im0rshdSSEIMqYNMyId2E/ZjfT5D3jPf4nXLKbe23BuOv2CX6u7C3ksNvD3iNmQpWwSfFkdhGU3mK6YoiYN0gneAYFS5+J2m3JoOUlsuLTCPjFR33Ob2dEgzxa1SWTXb9cerpniJg1DoVyO0I0OVxPnSTq4tl0bGI4dZUt1JZ1aePCJnsJRVw9cfrxy9JA5C75hy2wuUrpakXWs6iEYurUNKUr58H8DyWE8oSHTHtxs0Rj4YyxbBsy2SBWwtaN2z5yAT0kkDDRSP3iAbSWo5iFYuOYg0zOXwOlse64Ag0d0Xyoz8YixbBN8ILP6Ign6RQ/HhHWHEjBkQNdAKdZq0fbUcRCsXy/slC/Uz1s1dy2Pdi9E6SG5Q5t/bOoi2rBq+0jfFab9EDQehlKBmj6GGg5TIBeTrlVbROkiurTzDunUQbVk1PFdsWun/EjUchNLEmti6hoOUyAX+PwdpOsTi648W8+9WB7kgOL7WyK0OUipXmtZumd/MYtH3LGVV8DNELd5NIAcugQ6xlSjG4iAWuZZF+pGyWMC69pUaOT+rZikrZnuyt8ULSKVpXouRcrlaB7HKJQcpGayOlMUCHqMXyUzDT1NYyorh2/Wtxrwl+wID8kZ6wv6IUuIgNeWWJEuOlMUCHtcSqf7lgzhfQ1jKiqE9htIp/VXcIY/J6dj43ozTIYwoNR2kltwJbSZLNGh0lztlQFmo2IldS9kskuPtrXCFzGjISL8ROin13JCfMTWdXFNu64OVBI3u+F2a7bv8xG5s4LGUzbK9q9DybTaaTvcaSkYqjbP3BoQej5unewZbUy6l3Fvbi9Kg0R1BOuTXBnqsBxL3dGYpG2XEemlq+4xo11FyF6ZSbYo9E+JTbo8wLcfKpO7C15DL23jU9G6J7jjd8g7pk64RSAYLS9mPgabTGocWW4QuXrWYSXQOwhllO9OtQ/Hypzq/80I+8XekZrQb2joH5IzPcZJP+hE8pyFqHIl4Nz2OfXPQcRzHcRzHcRzHcRzHzl/mz/9+bAB3+AAAAABJRU5ErkJggg==" style="width: 100px; height: 19.5px;" width="100" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</div><div style="block-size: 58px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 29px; text-align: left; transform-origin: 384px 29px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 99.5px 8px; transform-origin: 99.5px 8px; unicode-bidi: normal; ">Given a polynomial function P<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 167px 8px; transform-origin: 167px 8px; unicode-bidi: normal; "> (presented as vector of numbers), and a value n<span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 10.5px 8px; transform-origin: 10.5px 8px; unicode-bidi: normal; ">, if <img src="data:image/png;base64,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" style="width: 91.5px; height: 19.5px;" width="91.5" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 29.5px 8px; transform-origin: 29.5px 8px; unicode-bidi: normal; ">, write a program that evaluates <img src="data:image/png;base64,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" style="width: 32px; height: 18.5px;" width="32" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 28px 8px; transform-origin: 28px 8px; unicode-bidi: normal; ">, where <img src="data:image/png;base64,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" style="width: 74px; height: 37px;" width="74" height="37"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; ">. </div><div style="block-size: 74.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 37.25px; text-align: left; transform-origin: 384px 37.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.5px 8px; transform-origin: 48.5px 8px; unicode-bidi: normal; ">For example, if <img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXsAAAAnCAYAAADuBegzAAAHGElEQVR4nO2d4bWqOhCFdw90YAM2YAVWYAd2YAe2YA2WYA+2YA22cO8P3IchBhLCBBKcby3We+t4rmiY2ZnsDBzAMAzDMAzDMAzDMAzDMAzDMAzDMAzDMDJyAHAZOIz6GLqWhzU/VAU0AG4AXgD+AXjDcsDYGBe0gf3wHEZ9+K6jCdc4DYAngCvaSfGMdsz+wfLA2BAXWEBvnQdM7Mc4oxV6yR6t2P/7/L/xQ+zQzvqhozZSxX6P9vvudD9OkfC7xhy5hGHO+9cu9vzuTab3v8Mfx0+0Yn/MdN45xOpRrbo0FdX826G/vONBC+QpfnZHmQHiY4rY09d0x+CFbQcUfdyYw60Q58Dxds/xRhuLsdQo9nu0eeR+9weWKzAeKLey98XF0PFc6TPmRis/Brk4byqrjQP6QqhywszEij19zTdaQbvgOxm3KPhHxCeV5hjs8D2ppk4stYn9Hu13f6HbYJbFlJt3uXh9jiXONYUG02Kypmsfi2Z+DHIVb3b3vH5CPyhLtzlixf6BdhZ1A5+JudXNLG5yjl1HJt9L+bwvtPFE9viuZGJWkDWJfYPhykzmVu7vs/uc5xT6xRW4IG41zQmydA1KQTM/BpEVhi8g3Vm3dDsnRuwP8E9shAP81vpQhbBDXGVHEdKycI5ox3LIPpCry7HrQmoS+wvGV8S01GK+9xxun3OUVtUD7RiErCVOVqVYOJqb3dr54cUV8qGTyd8p3cqJEfsjxqsDDm4pgaUFN8FC0M7SsnDuGI8bGYcxE2xNYh/KF/romnsjLge0sVxiRdwgroA8oywLR1PstfPDi/Rvx95ka5V9CFb2pU9sOWBgafrIN4SFhqK3NbEPwco+16YpV3MlCv0U6ECUso+mKfba+TF4En7o28DvyP7cGvyyuWJPj/WBMpe8uaGFMxQPuWAwx+63bEHsD5he1bNt1qXBd+twg3asShHIVGjhxO4h+caCDI3fVDTFPoYp+eFFtuANbdzIDpUpAjCll1uzn3aO2LNDp1Rvcwl4vZdewTEWY1ZTqWK/VkwOfZbYO4GPaK+L7NaQ1+eEflNBgy6WfdfxgjLbL4eghTM2KTaf1+UepCzYOPFpORRLi/2U/PgipmI/i9entmzJgU09UpYsKWLfoJ8wDJTSVzHa5LBwYpCVW8yYp4r9WjEp2aG/4UYRGxtvCgrHSW7UcXPv+vkvf/5Ev9WTxw31rVpjLJwGXexIx4JjxyJO616DJcV+an58IYX8jX7lckY/MVIq3Sv8zzWZcqTsPE8Ve3bm+ISghlZTTdaycCh+sVVLqtivFZPkhO8bFn1V6BiM0ye61YEbo6EblGrai5pq4QCdPUbH4gr9mF5S7KfmxxfSnnniO6ivaAeqNrGb69m7FX7uljig8xjnHFpdAUtbOHKPJJYtePYNviv8mGSW/2bIptFCw/aau4KIsXB8yDFK6aoLfXc5meS0/FLy4wsZZKV32ExBoxtH3s22hKUhgyf1mDsprWXh3NEm41SLsHaxJ7IjLiZuZazkLkQ0bK+5RUhqF4787KnPX1rb8gPS8qOHKy5bQuupl/LO4txLtT3m2wtzxW8NC+eMuBtqXLYk9kAnaLHiIL3+nGjYXnOcgRQLh8gVUIpQhr77mCuiYfkB6fnRQwqZhjD6qLEbRyInxJo6F1JhAC+1yjshPZC30I0joTDFiH2DrjMjV+6WAsclZVKTe5I5rllubZiTHz1Cj0jQoKZuHB8U+609MsEHK6ilLJy5gVxzN44PilpMJeg+rK+mrpqpcFKbKtZup2GOFVBOsVcTetm+lXNmqqUbZwjaGrmXyiXAKmgJC+eIcCDvMb78r7UbZwh2z4RWVVzWy1XnlvbbJBTsqRYOVz7sfKLVok0u/dTIjz8oYlutWrXE/gGl2bUCuNI7ZT4PWwVD/dKh57hsybOPvWOb4sffcavWBst0ji0FreapBcgd3Zj4fPszdCbIHGKvlR9/xDwioWZixJ6BxCrJJbbS2gLSwsnZZstAfmL4j4jzUbeh61eT2HMi9W1WMnF9RQVbM9na54qArFp3yN+GuTS0cELf6YjujmDeMEbcFskT9Kp8bbHXzI+/N5QWzhYfCxAj9u5NJxQP/lGJO36jogc6CyfnZp/8GwExRyjBaxJ7ipYssC5oC44Xhu+eDfXgnwOv1wx1KmYPSY6B7y7/d+D1VDTFXjs//kTQPW7Is1u9FjFi36C7S5jH9fOz2m4imwt97JxVIe9QjvXEQwlZk9jv8L1XwIo9ZNtw3HzXhn/CLve1WwP67THXmPnu+0NEQDs2Y6+noin22vnxM2h59ka51CT2xjahE/BrxWFRmNhvHxN7wzBM7H8AE3vDMP42WZe489HIj+86smPBMIwfxu1ikIdRH0PX0sTeMAzDMAzDMAzDMAzDMAzDMAzDMAzDMIxl+Q9RAimmlV7/YQAAAABJRU5ErkJggg==" style="width: 189.5px; height: 19.5px;" width="189.5" height="19.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 18px 8px; transform-origin: 18px 8px; unicode-bidi: normal; ">, and <img src="data:image/png;base64,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" style="width: 36.5px; height: 18px;" width="36.5" height="18"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 20px 8px; transform-origin: 20px 8px; unicode-bidi: normal; ">, then <img src="data:image/png;base64,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" style="width: 201.5px; height: 37px;" width="201.5" height="37"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; "> and <img src="data:image/png;base64,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" style="width: 186.5px; height: 37.5px;" width="186.5" height="37.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 37.5px 8px; transform-origin: 37.5px 8px; unicode-bidi: normal; ">. Therefore <img src="data:image/png;base64,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" style="width: 208.5px; height: 18.5px;" width="208.5" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; ">.</div><div style="block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; "><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; ">NOTE: <span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 51px 8px; transform-origin: 51px 8px; unicode-bidi: normal; ">It is possible for <img src="data:image/png;base64,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" style="width: 32px; height: 18.5px;" width="32" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 293.5px 8px; transform-origin: 293.5px 8px; unicode-bidi: normal; "> to return some complex numbers. We are interested only with real values, so in cases where there are no real <img src="data:image/png;base64,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" style="width: 32px; height: 18.5px;" width="32" height="18.5"><span style="block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 311px 8px; transform-origin: 311px 8px; unicode-bidi: normal; ">, please output an empty vector. Also please round-off your output to 4 decimal places, and sorted in ascending order.</div></div></div>

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William on 31 Oct 2021

I haven't quite figured out what problem 8 is supposed to do. I am getting a large number of real solutions, mostly 0.0 or 0.0001. I used unique() to reduce the solution to four values. These values are all correct except that they include 0.0.

Ramon Villamangca on 1 Nov 2021

Hi William, I missed to write "0.0000" in the solution set of test #8. Sorry for that. Corrected it.

Please rate the problem. Thanks.

Tim on 2 Nov 2021

For problem 8 I see only three real solutions--none that give R=0.0000.

Tim on 4 Nov 2021

I see now--you want to round off R before testing for an imaginary part.

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Problem 52973. Easy Sequences 45: Second Derivative of Inverse Polynomial Function

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Problem 52973. Easy Sequences 45: Second Derivative of Inverse Polynomial Function

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