Determine whether Rolle's Theorem can be applied to the function $f ( x ) = ( x + 3 ) ( x + 2 ) ^ { 2 }$ on the closed interval $( - 3 , - 2 )$. If Rolle's Theorem can be applied, find all numbers $c$ in the open interval $( - 3 , - 2 )$ such that $f ^ { t} ( c ) = 0$.
A) Rolle's Theorem applies; $- \frac { 8 } { 3 }$
B) Rolle's Theorem applies; $- \frac { 8 } { 5 }$
C) Rolle's Theorem applies; $- \frac { 4 } { 5 }$
D) Rolle's Theorem applies; $- \frac { 4 } { 3 }$
E) Rolle's Theorem does not apply

Question

Quizplus · Accepted Answer

Rolle's Theorem applies because $f(x)$ is continuous on the closed interval $[-3, -2]$ and differentiable on the open interval $(-3, -2)$, and $f(-3) = f(-2) = 0$. To find $c$, we differentiate $f(x)$ to get $f'(x) = 3x^2 + 12x + 8$ and solve $f'(c) = 0$. Factoring gives $3(x + 2)(x + \frac{4}{3}) = 0$, so $c = -\frac{4}{3}$ is the value in the interval $(-3, -2)$ where the derivative equals zero.

Determine Whether Rolle's Theorem Can Be Applied to the Function f(x)=(x+3)(x+2)2f ( x ) = ( x + 3 ) ( x + 2 ) ^ { 2 }f(x)=(x+3)(x+2)2

Determine Whether Rolle's Theorem Can Be Applied to the Function $f ( x ) = ( x + 3 ) ( x + 2 ) ^ { 2 }$