Find all critical numbers for the function. State whether it leads to a local maximum, a local minimum, or neither.
-$f(x)=-x^{3}-3 x^{2}+9 x+2$
A) Local maximum at 1 ; local minimum at -3
B) Local maximum at -3 ; local minimum at 1
C) Local maximum at -1 ; local minimum at 3
D) Local maximum at 3 ; local minimum at -1

Question

Quizplus · Accepted Answer

The critical numbers are -3 and 1. The second derivative is $f''(x)=-6x-6$. At $x=-3$, $f''(-3)=12>0$, so $f$ has a local minimum at $x=-3$. At $x=1$, $f''(1)=-12<0$, so $f$ has a local maximum at $x=1$.

Find All Critical Numbers for the Function f(x)=−x3−3x2+9x+2f(x)=-x^{3}-3 x^{2}+9 x+2f(x)=−x3−3x2+9x+2 A) Local Maximum at 1 ; Local Minimum at -3

Find All Critical Numbers for the Function $f(x)=-x^{3}-3 x^{2}+9 x+2$ A) Local Maximum at 1 ; Local Minimum at -3