Question

determine whether rolles theorem can be applied to ( f ) on the closed interval ( a, b ). (select all that apply.)
( f(x)=(x - 5)(x + 4)^{2}, quad-4,5 )
yes, rolles theorem can be applied.
no, because ( f ) is not continuous on the closed interval ( a, b ).
no, because ( f ) is not differentiable in the open interval ( (a, b) ).
no, because ( f(a)
eq f(b) ).
if rolles theorem can be applied, find all values of ( c ) in the open interval ( (a, b) ) such that ( f^{prime}(c)=0 ). (enter your answers as a comma-separated list. if rolles theorem cannot be applied, enter na.)
( c= )

Explanation:

Step1: Check continuity

A polynomial function \(y=(x - 5)(x + 4)^{2}=x^{3}+3x^{2}-24x - 80\) is continuous everywhere. So \(f(x)\) is continuous on \([-4,5]\).

Step2: Check differentiability

The derivative of \(y=(x - 5)(x + 4)^{2}\) using the product rule \((uv)^\prime=u^\prime v+uv^\prime\) where \(u=x - 5\), \(u^\prime=1\) and \(v=(x + 4)^{2}\), \(v^\prime = 2(x + 4)\). Then \(f^\prime(x)=(x + 4)^{2}+2(x - 5)(x + 4)=(x + 4)(x + 4+2x-10)=(x + 4)(3x - 6)\). A polynomial function is differentiable everywhere. So \(f(x)\) is differentiable on \((-4,5)\).

Step3: Check \(f(a)=f(b)\)

\(f(-4)=(-4 - 5)(-4 + 4)^{2}=0\) and \(f(5)=(5 - 5)(5 + 4)^{2}=0\). So \(f(-4)=f(5)\).
Since \(f(x)\) is continuous on \([-4,5]\), differentiable on \((-4,5)\) and \(f(-4)=f(5)\), Rolle's Theorem can be applied.

Step4: Find \(c\)

Set \(f^\prime(c)=0\). Since \(f^\prime(x)=(x + 4)(3x - 6)\), then \((c + 4)(3c - 6)=0\). Solving \(c+4 = 0\) gives \(c=-4\) (not in the open interval \((-4,5)\)), solving \(3c-6=0\) gives \(c = 2\).

Answer:

Yes, Rolle's Theorem can be applied. \(c = 2\)