Definition: Concavity describes how a graph bends: concave up when the slope is increasing, which means the second derivative is positive, and concave down when it is negative.
What is concavity?
A graph that bends like a cup is concave up; one that bends like a frown is concave down. The sign of the second derivative decides it: f” > 0 means concave up, and f” < 0 means concave down.
How do you find the intervals?
Find where f” is zero or undefined, then test the sign in each region.
Worked example: h(x) = 2x3 – 9x2 + 12x. Then h'(x) = 6x2 – 18x + 12 and h”(x) = 12x – 18. This is zero at x = 1.5. For x less than 1.5, h” is negative, so h is concave down. For x greater than 1.5, h” is positive, so h is concave up.
What mistake happens?
Confusing increasing with concave up. A function can be decreasing and still concave up, like the left half of a parabola opening upward. Mathbot separates the two questions: f’ for rising and falling, f” for bending.