find the absolute extrema if they exist, as well as all values of x where they occur, for the function (…

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 ).\nidentify 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.\na. the absolute maximum is ( -4.00 ), which occurs at ( x=-2.0 ).\n(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.)\nb. there is no absolute maximum.\nidentify 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.\na. the absolute minimum is ( -20.00 ), which occurs at ( x=-4 ).\n(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.)\nb. there is no absolute minimum.

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 ).\nidentify 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.\na. the absolute maximum is ( -4.00 ), which occurs at ( x=-2.0 ).\n(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.)\nb. there is no absolute maximum.\nidentify 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.\na. the absolute minimum is ( -20.00 ), which occurs at ( x=-4 ).\n(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.)\nb. there is no absolute minimum.

Answer

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).