What's the Relationship Between the Graphs of f, f', and f''?

The graph of (f') shows the gradient of $f$, while the graph of (f'') shows how that gradient changes. Therefore, (f') identifies increasing, decreasing, and stationary behaviour, while (f'') identifies concavity and helps classify stationary points.

This is IB Mathematics: Analysis and Approaches, SL 5.7: The second derivative, and it can be tested analytically on Paper 1 or using graphing technology on Paper 2.

Reading the graphs works in both directions. An x-intercept of (f') corresponds to a stationary point of $f$, while a turning point of (f') often occurs where $f''=0$. Likewise, the y-value of (f') at any $x$ equals the tangent gradient of $f$ there; a steeply rising has a large positive (f') value.

At a stationary point of $f$, the second-derivative test gives a quick classification: $f''(a)>0$ implies local minimum, while $f''(a)<0$ implies maximum. If , the test fails and you must inspect the sign of (f').

For example, let

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

Then

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

Since $f'(x)=0$ at $x=-1$ and $x=1$, these are stationary points. The sign of (f') changes from positive to negative at , giving a local maximum, and from negative to positive at , giving a local minimum. Since (f'') changes from negative to positive at , has a point of inflexion there.

A common misconception is that $f''(x)=0$ automatically proves a point of inflexion. It does not: the concavity must change, so exam answers should explicitly show a sign change in (f''). When interpreting graphs, connect each statement to evidence such as signs, intercepts, or turning points rather than relying only on appearance.