What's the Difference Between Concave Up and Concave Down?

A graph is concave up where its gradient is increasing, so $f''(x)>0$. It is concave down where its gradient is decreasing, so $f''(x)<0$.

How concavity works

The second derivative measures how the first derivative changes. Since (f'(x)) represents the gradient of the curve, the sign of (f''(x)) determines whether those gradients are increasing or decreasing.

Importantly, concavity is not the same as whether a function is increasing or decreasing. For example, a curve can be decreasing but concave up if its negative gradient is becoming less negative.

For example, $g(x)=e^{-x}$ is decreasing because $g'(x)=-e^{-x}<0$, but it is concave up because . The curve falls while flattening, showing why the signs of (f'(x)) and (f''(x)) answer different questions.

Worked example

Consider

$$f(x)=x^3-3x.$$

Differentiate twice:

$$f'(x)=3x^2-3,$$ $$f''(x)=6x.$$

A sign chart makes the conclusion explicit:

At $x=0$, the concavity changes. Therefore, $(0,0)$ is a point of inflexion.

A common misconception is that solving $f''(x)=0$ automatically proves a point of inflexion. It does not: you must confirm that (f''(x)) changes sign on either side of the point.

IB exam technique

This is HL-only content from AHL 5.10: Second derivatives, testing for maximum and minimum and may appear in any Math AI HL paper style. When asked to determine intervals of concavity, find (f''(x)), solve $f''(x)=0$, create a sign analysis, and state the intervals clearly. For a point of inflexion, show the change in concavity and give the full coordinate, not only the $x$-value. A GDC may check signs, but state the intervals mathematically.