3.3 · Derivative graphs, concavity, and curve analysis

The sign of the first derivative controls direction

Where $\displaystyle f'>0$, the function increases; where $\displaystyle f'<0$, it decreases. The vertical height of a derivative graph records slope, not the value of $\displaystyle f$. A derivative can be negative while increasing toward zero.

For $\displaystyle f'(x)=x^2-1$, signs are positive outside $\displaystyle [-1,1]$ and negative between. Thus $\displaystyle f$ has a local maximum at $\displaystyle -1$ and a local minimum at $\displaystyle 1$.

The second derivative controls changing slopes

Where $\displaystyle f''>0$, slopes increase and the graph is concave up. Where $\displaystyle f''<0$, slopes decrease and the graph is concave down. An inflection point requires continuity and a change in concavity.

Solving $\displaystyle f''=0$ gives candidates, not automatic inflection points. For $\displaystyle x^4$, the second derivative is zero at $\displaystyle 0$ but stays positive on both sides, so concavity does not change.

Use the second derivative test with its conditions

At a critical point with $\displaystyle f'(c)=0$, a positive second derivative gives a local minimum and a negative second derivative gives a local maximum. If $\displaystyle f''(c)=0$, this test is inconclusive; use sign analysis.

To sketch a curve, combine domain, intercepts, asymptotes, direction, and concavity. Evaluate $\displaystyle f$ to locate extrema as coordinate points, and keep any domain exclusions visible.

$$f'(c)=0,\ f''(c)>0\ \Rightarrow\ \mathrm{local\ minimum}$$