Increasing, decreasing, and extreme values of mathematical functions

Differentiation, derivatives and Taylor approximations: Applications of derivatives

Increasing, decreasing, and extreme values of mathematical functions

The behaviour of a function can be investigated using the derivative. The relationship between the graph of a smooth function $f(x)$ on an interval $I$ and derivative $f'(x)$ on the same interval is namely:

$\begin{aligned} \text{the graph of }f(x)\text{ is increasing on }I \iff f'(x)>0\\ \text{the graph of }f(x)\text{ is decreasing on }I \iff f'(x)<0\\ \text{the graph of }f(x)\text{ is stationary on }I \iff f'(x)=0\end{aligned}$

$f(a)$ is a global maximum (minimum) of the function $f$ when the function takes its largest (smallest) value in $a$ , i.e., $f(a)\ge f(x)$ for all $x$ (or in case of a local minimum $f(a)\le f(x)$ ). $f(a)$ is a local maximum (maximum) of the function $f$ when the function takes its largest (smallest) value in the neighbourhood of $a$ . The general term for maximum or minimum is extremum (plural: extrema) and the function values are also called extreme values.

If you want to find an extremum of a smooth function, you can first look at a point where the graph is stationary (also referred to as a critical point), that is, a point in the domain at which the tangent line on the graph is horizontal (derivative equal to zero). After all, each extremum is attained at a critical point because the derivative changes sign at this point. Once you have found a critical point, you must still determine whether it corresponds to an extremum and, if so, what kind of extremum it is. The following method applies:

If $f'(a)=0$ and the sign of $f'(x)$ changes at $x=a$ from positive to negative,
then $f(x)$ has a local maximum at $x=a$ .
If $f'(a)=0$ and the sign of $f'(x)$ changes at $x=a$ from negative to positive,
then $f(x)$ has a local minimum at $x=a$ .

The second derivative test for local extrema is a useful alternative.

The function $f(x)$ has a local maximum at $x=a$ if $f'(a)=0$ and $f''(a)<0$ .
The function $f(x)$ has a local minimum at $x=a$ if $f'(a)=0$ and $f''(a)>0$ .

If $f'(a)=0$ and $f''(a)=0$ , then you cannot conclude what kind of critical point $x=a$ is: it may correspond with an extremum, but it may also be a so-called inflection point, that is, a point where the derivative attains a maximum or minimum.

For the calculation of inflection points of a function you use the second and third derivative:

If $f''(a)=0$ and $f'''(a)\neq 0$ , then the function $f(x)$ has an inflection point at $x=a$ .

Mathcentre video

Maxima and Minima (38:19)