Definition: A critical point of a function is an input where the derivative equals zero or fails to exist, and it is a candidate for a local maximum or minimum.
What is a critical point?
It is a value c in the domain where f'(c) = 0 or f'(c) is undefined. Local extrema can occur only at critical points (or at endpoints of a closed interval).
How do you classify them?
Use the first derivative test (does f’ change sign?) or the second derivative test (sign of f”).
Worked example: f(x) = x3 – 3x2 – 24x + 5. Then f'(x) = 3x2 – 6x – 24 = 3(x – 4)(x + 2), so the critical points are x = -2 and x = 4. Since f”(x) = 6x – 6, we get f”(-2) = -18 (concave down) and f”(4) = 18 (concave up).
So there is a local maximum of 33 at x = -2 and a local minimum of -75 at x = 4.
What do students miss?
Assuming every critical point is an extremum. For y = x3, the derivative is zero at 0 but there is no max or min. Mathbot always tests the sign change.