Question

Let f be a decreasing function such that f(0) = 4 and f'(0)= 1 / 3. Which of the following is true about its inverse h? h'(0)= 1/3h ′(4) = 3h ′(4) = 0h'(1/3)= 4

Answer 1

Given:

$f(0)=4_{}$$f^{\prime}(0)=\frac{1}{3}$

We get the point (0,4) from f(0)=4.

we get the slope m=1/3 from f'(0)=1/3.

The point-slope formula is

$y-y_1=m(x-x_1)$$\text{ Substitute }y_1=4,x_1=0\text{ and m=}\frac{1}{3},\text{ we get}$

$y-4=\frac{1}{3}(x-0)$

$y-4=\frac{1}{3}x$$\text{Let y=f(x) and substitute x=f}^{-1}(y)$

$y-4=\frac{1}{3}f^{-1}(y)$

$3(y-4)=f^{-1}(y)$

Replace y by x, we get

$f^{-1}(x)=3(x-4)$$Let\text{ }f^{-1}(x)=h(x)$$h(x)=3(x-4)$

Differentiate with respect to x, we get

$h^{\prime}(x)=3$

Hence the answer is

$h^{\prime}(4)=3$