Question

Given the function `f(x)=-x^3+4x^2-3`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [0,4] and find the c?

Answer 1

The values of `c` are `={0,8/3}`

Explanation:

`f(x)` is a polynomial function.

So it is continuous on the interval `[0,4]`and differentiable on the interval `]0,4[`, therefore we can apply the mean value theorem which states that

there is `c in [0,4]` such that

`f'(c)=(f(4)-f(0))/(4-0)`

`f(x)=-x^3+4x^2-3`

`f(0)=-0+0-3=-3`

`f(4)=-64+64-3=-3`

Therefore,

`f'(c)=(f(4)-f(0))/(4-0)=(-3-(-3))/(4)=0`

Also,

`f'(x)=-3x^2+8x`

`f'(c)=-3c^2+8c=0`

Solving for `c`

`c(-3c+8)=0`

So,

`c=0`

and

`c=8/3`

Both values of `c in [0,4]`