Maxima and Minima

Maxima of a function. A function $f(x)$ is said to have a (local) maxima at $x=c$ if the value of the function at $x=c$ is greater than every other value of $f(x)$ in a small neighbourhood of $c$ — that is, for a small $\delta>0$ and for every $x\in(c-\delta,c+\delta)$, $f(c)>f(x)$. The value $f(c)$ is then called a maxima of $f(x)$. This happens exactly when $f(x)$ is increasing just to the left of $c$ (on $c-\delta<x<c$) and decreasing just to the right of $c$ (on $c<x<c+\delta$).

Minima of a function. Similarly, $f(x)$ has a (local) minima at $x=c$ if $f(c)<f(x)$ for every $x$ in a small neighbourhood of $c$. This happens when $f(x)$ is decreasing just to the left of $c$ and increasing just to the right of $c$.

Stationary (turning) points. If $f'(c)=0$, then at $x=c$ the function is neither increasing nor decreasing at that instant — such a point is called a stationary point or turning point of the function. Geometrically, any point where the tangent to the graph is horizontal is a turning point, so turning points are located by solving $\dfrac{dy}{dx}=0$. Among the turning points, the ones where $f$ genuinely attains a local maximum or minimum are called the extreme values of the function. …