Applications of Derivatives — Part 2: Maxima, minima and optimisation: visual lesson, ISC Class 12 Mathematics

Lesson

Applications of Derivatives, Part 2

The largest pen, the cheapest tin, the best page layout: optimisation with derivatives.

In this lesson you will

Find critical points and classify them by the first derivative test
Use the second derivative test and recognise when it fails
Find absolute maximum and minimum values on a closed interval
Set up and solve optimisation problems from geometry and daily life

Part 2 at a glance.

Concept

Critical and stationary points

$c$ is a critical point if $f'(c) = 0$ or $f'(c)$ does not exist. A stationary point has $f'(c) = 0$ . Local extrema occur only at critical points.

$y = -x^3 + 3x^2 + 9x - 4$

A local maximum is the top of a hill, not necessarily the highest point overall.

Concept

The first derivative test

Watch the sign of $f'$ as $x$ passes the critical point $c$ .

Look at the sign of the derivative just before and just after $c$.
Sign of $f'$ as $x$ passes $c$	Conclusion at $c$
$+$ to $-$	local maximum
$-$ to $+$	local minimum
no change	neither (for example a point of inflection)

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