Algebra.Com's Answer #745677 by ikleyn(52787)![\"\"](\"/images/stars/stars-13.gif\")  
You can put this solution on YOUR website!
.
\n" );
document.write( "Use the function f and the given real number a to find (f^ −1)'(a).\r
\n" );
document.write( "\n" );
document.write( "f(x) = x3 + 5x − 1, a = −7
\n" );
document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "\r\n" );
document.write( "The problem asks you to find the value of the DERIVATIVE of the inverse function g(y) to the given function \r\n" );
document.write( "\r\n" );
document.write( " y = f(x) = x^3 + 5x -1\r\n" );
document.write( "\r\n" );
document.write( "at the point y= f(x) = -7.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The key idea in solving the problem is to use WELL KNOWN identity\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "  = 1. (1)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Our first step is to determine the value of \"x\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "For it, we first solve the equation\r\n" );
document.write( "\r\n" );
document.write( " x^3 + 5x - 1 = -7, (2)\r\n" );
document.write( "\r\n" );
document.write( "which is equivalent to\r\n" );
document.write( "\r\n" );
document.write( " x^3 + 5x + 6 = 0. (3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It easy to guess and then to check that x= -1 is the solution.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " Then performing long division or synthetic division of the given polynomial by (x+1), you find the second polynomial factor,\r\n" );
document.write( " which has complex roots; so, the equation (2) has UNIQUE real solution x= -1.\r\n" );
document.write( "\r\n" );
document.write( " I do not go into details here, since it is only an auxiliary melody - not the main theme.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus we know that x= -1 is the solution to (2), and we easily can calculate the derivative  at this point: it is\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " =  at x= -1, which is 3*(-1)^2+5 = 3+5 = 8.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Then, according to (1), for the inverse function g(y) to function f(x), we have\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "  =  =  . ANSWER\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is what the problem asks to get.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER. (f^(-1))'(-7) = .\r\n" );
document.write( " \r
\n" );
document.write( "\n" );
document.write( "Solved.\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "-----------------\r
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( "Tutor @greenestamps misread the problem, so his answer and his solution are IRRELEVANT.\r
\n" );
document.write( "
\n" );
document.write( "
\n" );
document.write( "\n" );
document.write( " \n" );
document.write( " |