Determine all critical points for the function.
-$$f ( x ) = 80 x ^ { 3 } - 3 x ^ { 5 }$$
A) x = -4
B) x = -4 and x = 4
C) x = 0, x = -4, and x = 4
D) x = 4

Question

Quizplus · Accepted Answer

To find the critical points, we need to find the derivative of the function:

$$f'(x) = 240x^2 - 15x^4$$

Now, we need to set this equal to zero and solve for x:

$$240x^2 - 15x^4 = 0$$

Factoring out 15x^2, we get:

$$15x^2(16 - x^2) = 0$$

Using the zero product property, we get:

$$15x^2 = 0 \quad \text{or} \quad 16 - x^2 = 0$$

Solving for x in each equation:

$$x = 0 \quad \text{or} \quad x = \pm 4$$

These are the critical points of the function. Therefore, the answer is choice C, as it includes all three critical points.

Determine All Critical Points for the Function f(x)=80x3−3x5f ( x ) = 80 x ^ { 3 } - 3 x ^ { 5 }f(x)=80x3−3x5

Determine All Critical Points for the Function $f ( x ) = 80 x ^ { 3 } - 3 x ^ { 5 }$