Question

an increasing function f satisfies f(10)=5 and f(10)=8. which of the following statements about the inverse of f must be true? (a) (f^(-1))(5)=10 (b) (f^(-1))(8)=10 (c) (f^(-1))(5)=8 (d) (f^(-1))(5)=1/8

Explanation:

Step1: Recall inverse - function derivative formula

The formula for the derivative of the inverse function is \((f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}\). We know that \(f(5) = 10\), so \(f^{-1}(10)=5\), and \(f'(10)=8\).

Step2: Apply the formula

Substitute \(a = 10\) into the formula \((f^{-1})'(a)=\frac{1}{f'(f^{-1}(a))}\). We get \((f^{-1})'(10)=\frac{1}{f'(f^{-1}(10))}\). Since \(f^{-1}(10) = 5\) and \(f'(10)=8\), then \((f^{-1})'(10)=\frac{1}{8}\).

Answer:

D. \((f^{-1})'(10)=\frac{1}{8}\)