AP Calculus AB and BC glossary

Concave down

Also called: Concave downward

A graph is concave down on an interval where the second derivative is negative. The slope is decreasing, every tangent line lies above the curve, and a tangent line estimate there is an overestimate. Inflection points separate concave down from concave up.

Concave down constrains only the second derivative. The function and its slope can be positive or negative, so a curve can be rising and concave down at the same time, which is the shape of a quantity that grows but decelerates, like $\sqrt{x}$ for $x > 0$ .

Because the graph sits below its tangent lines here, a tangent line approximation overestimates the true values, the mirror image of the concave up case.

The mistake

Reading $f''(x) < 0$ as $f'(x) < 0$ or $f(x) < 0$ . Concave down means the slope is decreasing and says nothing about whether the function or its slope is positive. A steadily rising curve can be concave down the whole way.

Appears in: Unit 5: Analytical Applications