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?

1 Answer
Jan 26, 2017

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]#