Question

let fand g be inverse functions that are differentiable for all x. if f(3)= -2 and g(-2)=-4, which of the following statements must be false? 1.F’(0)=¼ 2.F’(3)=¼ 3.F’(5)=¼

Answer 1

Statement 3, F'(5) = 1/4, must be false.

How to solve

Inverse functions and derivative relationship: If f and g are inverse functions, then:

f(g(x)) = x for all x

g(f(x)) = x for all x

Differentiate both equations: Differentiating both equations with respect to x, we get:

f'(g(x)) * g'(x) = 1 (chain rule)

g'(f(x)) * f'(x) = 1 (chain rule)

Plug in given values:

We are given f(3) = -2 and g(-2) = -4.

Using f(3) = -2 in the first equation: f'(g(-2)) * g'(-2) = 1.

Using g(-2) = -4 in the second equation: g'(f(3)) * f'(3) = 1.

Solve for F'(3):

From the first equation, f'(g(-2)) = 1 / g'(-2) = 1 / (-4) = -1/4.

Substitute this into the equation from step 3: g'(f(3)) * (-1/4) = 1.

Therefore, g'(f(3)) = -4.

Analyze statements:

We don't have enough information to determine the truth of statements 1 (F'(0) = 1/4) and 2 (F'(3) = 1/4).

However, statement 3 (F'(5) = 1/4) contradicts our analysis. Since g'(f(3)) = -4, regardless of the value of f(5), f'(5) cannot be 1/4 for the second equation to hold true.

Therefore, statement 3, F'(5) = 1/4, must be false.