Identify the function's local and absolute extreme values, if any, saying where they occur.
-$$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 4 x - 3$$
A) local maximum at $x = - 2$; local minimum at $x = - \frac { 2 } { 3 }$
B) local maximum at $x = - \frac { 2 } { 3 }$; local minimum at $x = - 2$
C) local maximum at $x = 2$; local minimum at $x = \frac { 2 } { 3 }$
D) local maximum at $x = \frac { 2 } { 3 }$; local minimum at $x = 2$

Question

Quizplus · Accepted Answer

To find the local extrema, we first find the derivative of the function $f(x) = x^3 + 4x^2 + 4x - 3$, which is $f'(x) = 3x^2 + 8x + 4$. Setting $f'(x) = 0$ gives the critical points. Solving $3x^2 + 8x + 4 = 0$ using the quadratic formula yields $x = -2$ and $x = -\frac{2}{3}$. By testing intervals around these critical points or using the second derivative test, we can determine that $x = -2$ is a local maximum and $x = -\frac{2}{3}$ is a local minimum.

Identify the Function's Local and Absolute Extreme Values, If Any f(x)=x3+4x2+4x−3f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 4 x - 3f(x)=x3+4x2+4x−3

Identify the Function's Local and Absolute Extreme Values, If Any $f ( x ) = x ^ { 3 } + 4 x ^ { 2 } + 4 x - 3$