Need help of max and min calculus, thank you

Question

Find the x-coordinate of the point(s) of inflection on the graph of y = x³ - 12x² - 5.

Options:

1. x = -2
2. x = 4
3. x = 2
4. x = 12

Answer 1

Hello Marietta,

An inflection point for a function f is a point on the graph of f where the concavity changes. First, let us find the intervals where f is concave up (or concave down.) For this, we use the sign of the second derivative. We have

f´(x) = 3x² - 24x, and

f ´´(x) = 6x - 24.

The function will be concave up when f´´(x) > 0, so we solve the resulting inequality:

6x - 24 > 0

6x > 24

x > 4.

Thus, the graph of f(x) is concave up on the interval (4,∞).

Similarly, f is concave down when f´´(x) < 0, so solving 6x - 24 < 0 gives x < 4. Therefore, f is concave down on the interval (-∞,4). Since the function changes from concave up to concave down at x = 4, the function has an inflection point at x = 4. (Choice 2.)

Hope that helps! Let me know if you need any clarification.

William