Over- and Under-Estimates from Concavity

A tangent line only matches a curve exactly at the point of tangency. Everywhere else, the curve either pulls above the line or sags below it, and which way it pulls is exactly what concavity measures. If a curve is concave down, it bends downward away from the tangent line on both sides, so the tangent line sits above the curve, and using it to estimate a nearby value gives something too big. If a curve is concave up, it bends upward away from the tangent line, so the line sits below the curve, and the estimate comes out too small.

The mechanism is the second derivative. f''(x) < 0 means concave down, tangent line overestimates. f''(x) > 0 means concave up, tangent line underestimates. This only tells you the sign of the error, not how large it is. A tangent line can underestimate by a tiny sliver or by a huge margin depending on how sharply the function curves and how far you step from the point of tangency, so 'underestimate' is not automatically 'less wrong' than 'overestimate'.

This sits right after linear approximation in the unit: first you learn to build the tangent line estimate L(x) = f(a) + f'(a)(x - a), then this topic tells you whether to trust it as too high or too low. It also sets up a subtlety worth remembering for later: at a point of inflection, concavity switches sign right at that point, so the tangent line estimate is reliable in neither direction nearby and the overestimate/underestimate story breaks down.