Question

Given the function `f(x)=x^3-9x^2+24x-18`, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [2,4] and find the c?

Answer 1

The values of `c` are `{2.42, 3.58}`

Explanation:

`f(x)` is a polynomial function.

So it is continuous on the interval `[2,4]`and differentiable on the interval `]2,4[`, therefore we can apply the mean value theorem which states that

there is `c in [2,4]` such that

`f'(c)=(f(4)-f(2))/(4-2)`

`f(x)=x^3-9x^2+24x-18`

`f(2)=8-36+48-18=2`

`f(4)=64-144+96-18=-2`

Therefore,

`f'(c)=(f(4)-f(2))/(4-2)=(-2-2)/(4-2)=-2`

Also,

`f'(x)=3x^2-18x+24`

`f'(c)=3c^2-18c+24=-2`

Solving for `c`

`3c^2-18c+26=0`

Solving for `c`

`Delta=18^2-4*3*26=12`

`Delta>0`, there are 2 real roots

`c=(18+-sqrtDelta)/6=(18+-sqrt12)/6=(9+-sqrt3)/3`

So,

`c=(9+sqrt3)/3=3.58`

and

`c=(9-sqrt3)/3=2.42`

Both values of `c in [2,4]`