Applications of Differentiation

Maxima & minima

Many of the problems you will come across that apply differentiation involve optimisation of some kind. This means you are trying to find the maximums or minimums of a function. You may recall studying stationary points in high school.

"Stationary points occour at $f'(x) = 0$ " - Ms Fyfe 2K16

The values of $x$ such that $f'(x) = 0$ are called critical points. However, there are two different types of maximums and minimums found in functions:

Local maximum / minimum
A value is considered a local critical point if it is greater than or less than other values of $f(x)$ near that $x$ .
Absolute maximum / minimum
A value is considered an absolute critical point if it is greater than or less than other values of $f(x)$ for all $x$ in the domain.

Note that absolute maxima & minima are also technically local maxima & minima in restricted bounds.

Extreme value theorem

If $f$ is continuous on a closed interval $[a,b]$ , then $f$ attains an absolute maximum value $f(c)$ and an absolute minimum value $f(d)$ at some numbers $c$ and $d$ in the interval $[a,b]$ .

Fermat’s theorem

If $f$ has a local maximum or minimum at $c$ , and $f'(c)$ exists, then $f'(c) = 0$ .

This is just the formal definition of what is said at the top of this topic.

Critical numbers

A critical number of a function $f$ is a number $c$ in the domain of $f$ such that either $f'(c) = 0$ or $f'(c)$ does not exist.

Finding the critical numbers of a function is like finding the roots of the derivative.

Rolle’s theorem

Let $f$ be a function that satisfies the following three hypotheses:

$f$ is continuous on the closed interval $[a,b]$
$f$ is differentiable on the open interval $(a,b)$
$f(a) = f(b)$

Then there is a number $c$ in $(a,b)$ such that $f'(c) = 0$ .

Mean value theorem

The mean value theorem states that if a function $f$ is continuous over $[a,b]$ and differentiable over $(a,b)$ then there exists a number $c$ in $(a,b)$ such that:

$f'(c) = \dfrac{f(b) - f(a)}{b - a}$

This is essentially saying that you can draw a tangent line at point $c$ that is parallel to a straight line drawn between points at $a$ and $b$ .

Also check out the extreme value theorem or intermediate value theorem when asked about any proof for roots, critical points or concavity. Sometimes proof just needs to be explained without any actual working.