Question 1128664
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Use the function f and the given real number a to find (f^ −1)'(a).

f(x) = x3 + 5x − 1,    a = −7
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<pre>
The problem asks you to find the value of the <U>DERIVATIVE</U> of the inverse function g(y) to the given function 

    y = f(x) = x^3 + 5x -1

at the point  y= f(x) = -7.



The <U>key idea</U> in solving the problem is to use <U>WELL KNOWN identity</U>


    {{{((dg(y))/(dy))*((df(x))/(dx))}}} = 1.     (1)



Our first step is to determine the value of "x".


For it, we first solve the equation

    x^3 + 5x - 1 = -7,      (2)

which is equivalent to

    x^3 + 5x + 6 = 0.       (3)


It easy to guess and then to check that  x= -1 is the solution.


    Then performing long division or synthetic division of the given polynomial by (x+1), you find the second polynomial factor,
    which has complex roots; so, the equation  (2)  has UNIQUE real solution x= -1.

    I do not go into details here, since it is only an auxiliary melody - not the main theme.


Thus we know that  x= -1 is the solution to  (2),  and we easily can calculate the derivative  {{{(df(x))/(dx)}}} at this point: it is


     {{{(df(x))/(dx)}}} = {{{(3x^2 +5)}}} at x= -1,  which is  3*(-1)^2+5 = 3+5 = 8.


Then, according to (1), for the inverse function g(y) to function f(x), we have


     {{{(dg(-7))/(dy)}}} = {{{1/((df(-1))/(dx))}}} = {{{1/8}}}.         <U>ANSWER</U>


It is what the problem asks to get.


<U>ANSWER</U>.  (f^(-1))'(-7) = {{{1/8}}}.
</pre>

Solved.


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Tutor @greenestamps misread the problem, so his answer and his solution are IRRELEVANT.