Determine the critical points of the given function and classify each critical point as a relative maximum, a relative minimum, or neither. $f ( x ) = 6 x ^ { 4 } - 48 x ^ { 3 } + 108 x ^ { 2 } + 4$
A) (0, 4) relative minimum; (3, 166) neither
B) (0, 2) neither; (1, 3) relative minimum
C) (0, 2) relative minimum; (1, 4) neither
D) (0, 2) relative minimum; (1, 3) relative maximum

Question

Quizplus · Accepted Answer

To find the critical points of the function $f(x) = 6x^4 - 48x^3 + 108x^2 + 4$, we first find its derivative: $f'(x) = 24x^3 - 144x^2 + 216x$. Setting $f'(x) = 0$ gives us the critical points. Factoring out $24x$, we get $24x(x^2 - 6x + 9) = 24x(x-3)^2 = 0$, which gives us $x = 0$ and $x = 3$ as critical points. To classify these points, we examine the second derivative, $f''(x) = 72x^2 - 288x + 216$. Evaluating $f''(0)$ gives a positive value, indicating a relative minimum at $x = 0$. Evaluating $f''(3)$ gives $0$, which means the second derivative test is inconclusive at $x = 3$; however, by examining the behavior of the function or further analysis, we can determine the nature of the critical point at $x = 3$. The correct classification is that $x = 0$ is a relative minimum, and the nature of the critical point at $x = 3$ can be further analyzed, but it is typically considered neither a relative maximum nor a relative minimum without additional context. The exact nature of the critical point at $x = 3$ might require more detailed analysis beyond the second derivative test, such as examining the sign changes of the first derivative around $x = 3$. However, the provided options do not fully align with a detailed mathematical analysis, suggesting a potential simplification or error in the interpretation of the critical point at $x = 3$. Given the options, (A) is the closest to a correct interpretation based on the initial analysis.

Determine the Critical Points of the Given Function and Classify f(x)=6x4−48x3+108x2+4f ( x ) = 6 x ^ { 4 } - 48 x ^ { 3 } + 108 x ^ { 2 } + 4f(x)=6x4−48x3+108x2+4

Determine the Critical Points of the Given Function and Classify $f ( x ) = 6 x ^ { 4 } - 48 x ^ { 3 } + 108 x ^ { 2 } + 4$