Question

find the absolute extrema if they exist, as well as all values of x where they occur, for the function ( f(x)=x^{3}+4 x^{2}+4 x - 4 ) on the domain ( -4,0 ).
identify the absolute maximum if it exists, as well as all values of ( x ) where it occurs. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute maximum is ( -4.00 ), which occurs at ( x=-2.0 ).
(round the absolute maximum to two decimal places as needed. type an exact answer for the value of ( x ) where the maximum occurs. use a comma to separate answers as needed.)
b. there is no absolute maximum.
identify the absolute minimum if it exists, as well as all values of ( x ) where it occurs. select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
a. the absolute minimum is ( -20.00 ), which occurs at ( x=-4 ).
(round the absolute minimum to two decimal places as needed. type an exact answer for the value of ( x ) where the minimum occurs. use a comma to separate answers as needed.)
b. there is no absolute minimum.

Explanation:

Step1: Find the derivative of the function

The function is \(f(x)=x^{3}+4x^{2}+4x - 4\).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \(f^\prime(x)=3x^{2}+8x + 4\).
Factor \(f^\prime(x)\): \(f^\prime(x)=(3x + 2)(x+2)\).
Set \(f^\prime(x)=0\), then \(3x + 2 = 0\) gives \(x=-\frac{2}{3}\), and \(x + 2=0\) gives \(x=-2\).

Step2: Evaluate the function at critical points and endpoints

The domain is \([-4,0]\).
Evaluate \(f(x)\) at \(x=-4\): \(f(-4)=(-4)^{3}+4(-4)^{2}+4(-4)-4=-64 + 64-16 - 4=-20\).
Evaluate \(f(x)\) at \(x=-2\): \(f(-2)=(-2)^{3}+4(-2)^{2}+4(-2)-4=-8 + 16-8 - 4=-4\).
Evaluate \(f(x)\) at \(x =-\frac{2}{3}\): \(f(-\frac{2}{3})=(-\frac{2}{3})^{3}+4(-\frac{2}{3})^{2}+4(-\frac{2}{3})-4=-\frac{8}{27}+\frac{16}{9}-\frac{8}{3}-4=-\frac{8 + 48-72 - 108}{27}=-\frac{140}{27}\approx - 5.19\).
Evaluate \(f(x)\) at \(x = 0\): \(f(0)=0^{3}+4\times0^{2}+4\times0-4=-4\).

Answer:

A. The absolute maximum is \(-4.00\), which occurs at \(x=-2.0\) and \(x = 0\).
A. The absolute minimum is \(-20.00\), which occurs at \(x=-4\).