25: a function f, with a differentiable inverse function f, is such that f(1) = 1/3, f(1) = 2 and f(2) = 0…

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).\n(a) 2 (b) 1 (c) 1/3 (d) 3 (e) none of these.\n26: 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).\n(a) -2 (b) -1 (c) 1 (d) 2 (e) none of these.

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).\n(a) 2 (b) 1 (c) 1/3 (d) 3 (e) none of these.\n26: 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).\n(a) -2 (b) -1 (c) 1 (d) 2 (e) none of these.

Answer

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