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?

1 Answer
Jan 26, 2017

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]#