Find the exact location of all the absolute extrema of the function with domain $( - 7 , \infty )$ . ? $f ( x ) = x ^ { 4 } - 8 x ^ { 3 }$
A) $( 6,432 )$ - absolute minimum
B) $( - 6 , - 432 )$ - relative maximum
C) $( 6,0 )$ - absolute maximum
D) $( 6 , - 432 )$ - absolute minimum
E) $( 6 , - 432 )$ - relative minimum

Question

Find the exact location of all the absolute extrema of the function with domain $( - 7 , \infty )$ . ? $f ( x ) = x ^ { 4 } - 8 x ^ { 3 }$ 
A) $( 6,432 )$ - absolute minimum
B) $( - 6 , - 432 )$ - relative maximum
C) $( 6,0 )$ - absolute maximum
D) $( 6 , - 432 )$ - absolute minimum
E) $( 6 , - 432 )$ - relative minimum

Quizplus · Accepted Answer

To find the absolute extrema, we first find the derivative of $f(x) = x^4 - 8x^3$, which is $f'(x) = 4x^3 - 24x^2$. Setting $f'(x) = 0$ gives us the critical points $x = 0$ and $x = 6$. Since the domain is $(-7, \infty)$, we only consider $x = 6$. Plugging $x = 6$ into $f(x)$ gives $f(6) = 6^4 - 8(6)^3 = 1296 - 1728 = -432$, indicating an extremum at $(6, -432)$. To determine if it's an absolute minimum, we consider the behavior of $f(x)$ as $x$ approaches the endpoints of the domain and at critical points. Since $f(x)$ tends to infinity as $x$ goes to infinity and there are no other critical points within the domain that yield a lower value, $(6, -432)$ is indeed an absolute minimum.

Find the Exact Location of All the Absolute Extrema of the Function