Find all relative extrema of the function $f ( x ) = 2 x ^ { 4 } - 32 x ^ { 3 } + 4$ . Use the Second Derivative Test where applicable.
A) relative max: $( 24,221188 )$; no relative min
B) relative min: $( 12 , - 13820 )$; no relative max
C) relative min: $( 24,221188 )$; no relative max
D) relative max: $( 12,13820 )$; no relative min
E) no relative max or min

Question

Find all relative extrema of the function $f ( x ) = 2 x ^ { 4 } - 32 x ^ { 3 } + 4$ . Use the Second Derivative Test where applicable. 
A) relative max: $( 24,221188 )$; no relative min
B) relative min: $( 12 , - 13820 )$; no relative max
C) relative min: $( 24,221188 )$; no relative max
D) relative max: $( 12,13820 )$; no relative min
E) no relative max or min

Quizplus · Accepted Answer

To find the relative extrema of the function $f(x) = 2x^4 - 32x^3 + 4$, we first find its first derivative $f'(x) = 8x^3 - 96x^2$ and set it equal to zero to find critical points: $8x^3 - 96x^2 = 0$. Factoring out $8x^2$, we get $8x^2(x - 12) = 0$, which gives us $x = 0$ and $x = 12$ as critical points. The second derivative $f''(x) = 24x^2 - 192x$ is used to apply the Second Derivative Test. For $x = 12$, $f''(12) = 24(12)^2 - 192(12) = 3456 - 2304 = 1152 > 0$, indicating a relative minimum at $x = 12$. Substituting $x = 12$ into the original function to find the y-coordinate of the extremum, $f(12) = 2(12)^4 - 32(12)^3 + 4 = -13820$. Therefore, there is a relative minimum at $(12, -13820)$, with no relative maximum identified through this process.

Find All Relative Extrema of the Function f(x)=2x4−32x3+4f ( x ) = 2 x ^ { 4 } - 32 x ^ { 3 } + 4f(x)=2x4−32x3+4

Find All Relative Extrema of the Function $f ( x ) = 2 x ^ { 4 } - 32 x ^ { 3 } + 4$