Question

Find all global extrema for f(x) = x³ - 3 on the interval [−3,5).

a) Global maximum at x=−3, Global minimum at x=5
b) Global maximum at x=5, Global minimum at x=−3
c) Global maximum at x=0, Global minimum at x=5
d) Global maximum at x=−3, Global minimum at x=0

Answer 1

The global maximum occurs at x = 5 and the global minimum occurs at x = -3.

Step-by-step explanation:

The global extrema of a function occur at the highest and lowest points of the graph. To find the global extrema of f(x) = x³ - 3 on the interval [-3,5), we can evaluate the function at the endpoints and at the critical points within the interval.

To find critical points, we take the derivative of f(x) and set it equal to zero.

Differentiating f(x) = x³ - 3, we get f'(x) = 3x². Setting this equal to zero, we find the critical point at x = 0.

Evaluating f(x) at the endpoints and the critical point, we find that f(-3) = -24, f(0) = -3, and f(5) = 122.

Therefore, the global maximum occurs at x = 5 and the global minimum occurs at x = -3.