Question

Let f be the function given by f(x) = x³ - 6x². The graph of f is concave up when ______.

Answer 1

The graph of the function f(x) = x³ - 6x² is concave up where the second derivative f''(x) is positive, which occurs when x > 2.

Step-by-step explanation:

The graph of the function f(x) = x³ - 6x² is concave up when its second derivative is positive. To find where the function is concave up, we compute the second derivative:

1. First derivative: f'(x) = 3x² - 12x.
2. Second derivative: f''(x) = 6x - 12.

To determine where f''(x) > 0, we set the second derivative greater than zero and solve for x:

6x - 12 > 0

6x > 12
x > 2

Thus, the graph of f is concave up when x > 2.