Final answer:
The function y = f(x) = x³ − 12x is concave downward for x < 0 and concave upward for x > 0. The point (0, 0) is an inflection point where the concavity changes.
Step-by-step explanation:
To determine where the function y = f(x) = x³ − 12x is concave up or concave down, we need to consider the second derivative of the function. The second derivative tells us about the concavity of the function. The first derivative is f'(x) = 3x² - 12, and taking the derivative again gives us f''(x) = 6x.
For the function to be concave upward, f''(x) must be positive, and for it to be concave downward, f''(x) must be negative. Setting f''(x) = 0, we find x = 0 which is a potential inflection point where concavity could change.
Examining f''(x) on either side of x = 0, we see that for x < 0, f''(x) is negative (concave downward), and for x > 0, f''(x) is positive (concave upward). So, the function is concave downward for x < 0 and concave upward for x > 0. The point (0, f(0)), which is (0, 0), is an inflection point because the concavity changes from down to up as x increases through 0.