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)=(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=}(1)/(3),\text{ we get}`

`y-4=(1)/(3)(x-0)`

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

`y-4=(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`