AP Calculus AB and BC glossary
Concave up
A graph is concave up on an interval where the second derivative is positive. The slope is increasing, tangent lines lie below the curve, and any tangent line approximation there is an underestimate.
Concave up says nothing about whether the function is rising or falling. A decreasing function that is concave up is falling ever more slowly, which is the shape of exponential decay.
The mistake
Assuming concave up means the function has a minimum there. It only means the slope is increasing; whether a minimum exists depends on the first derivative changing sign.
Appears in: Unit 5: Analytical Applications