Question

25: a function f, with a differentiable inverse function f, is such that f(1) = 1/3, f(1) = 2 and f(2) = 0. find f(2).
(a) 2 (b) 1 (c) 1/3 (d) 3 (e) none of these.
26: a function f, with a differentiable inverse function f, is such that f(0) = -1, f(1) = 0 and f(0) = 2. find f(2).
(a) -2 (b) -1 (c) 1 (d) 2 (e) none of these.

Explanation:

Step1: Recall inverse - function derivative formula

If \(F\) is the inverse of a differentiable function \(f\), then \(F^{\prime}(y)=\frac{1}{f^{\prime}(x)}\), where \(y = f(x)\).

Step2: For question 25

We want to find \(F^{\prime}(2)\). Since \(F\) is the inverse of \(f\) and \(f(1)=2\), then by the formula \(F^{\prime}(2)=\frac{1}{f^{\prime}(1)}\). Given \(f^{\prime}(1)=\frac{1}{3}\), so \(F^{\prime}(2)=\frac{1}{\frac{1}{3}} = 3\).

Step3: For question 26

We want to find \(F^{\prime}(2)\). Since \(F\) is the inverse of \(f\) and \(f(0)=2\), then by the formula \(F^{\prime}(2)=\frac{1}{f^{\prime}(0)}\). Given \(f^{\prime}(0)=- 1\), so \(F^{\prime}(2)=\frac{1}{-1}=-1\).

Answer:

1. d. 3
2. b. - 1