Question

In the given function f(x)=x⁴ - 8x², the values of x are restricted within the interval [−3,1]. What is are the critical point(s) within this interval?

a) x = -3, x = 1
b) x = -2, x = 2
c) x = -1, x = 0
d) x = -3, x = 0, x = 1

Answer 1

The critical points within the interval [−3,1] for the function f(x)=x⁴ - 8x² are x = -2, x = 0, and the interval endpoints x = -3 and x = 1.

Step-by-step explanation:

To find the critical points within the interval [−3,1] for the function f(x)=x⁴ - 8x², we first need to find the derivative of the function f'(x) and determine where it equals zero or is undefined. The derivative of f(x) is f'(x) = 4x³ - 16x. Setting the derivative equal to zero gives us the equation 4x³ - 16x = 0. Factoring out 4x yields 4x(x² - 4) = 4x(x - 2)(x + 2) = 0. Within the interval [−3,1], the values that make the derivative zero are x = -2 and x = 0, as x = 2 is outside the interval. Additionally, the endpoints of the interval, x = -3 and x = 1, are always considered critical points because they are the boundaries of the closed interval.