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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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The problem asks you to find the value of the DERIVATIVE 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 key idea in solving the problem is to use WELL KNOWN identity
= 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 at this point: it is
= 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
= = . ANSWER
It is what the problem asks to get.
ANSWER. (f^(-1))'(-7) = .
Solved.
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Tutor @greenestamps misread the problem, so his answer and his solution are IRRELEVANT.