Maxima, Minima and Optimisation

Name: Maxima, Minima and Optimisation
Author: StemPy Lab

Where does a graph turn? Find the critical points of a function and decide whether each is a maximum, a minimum or neither, with the first and the second derivative tests. Find the largest and smallest values on a closed interval. Then solve optimisation problems: the biggest box, the cheapest tin, the shortest fence. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Critical points and the first derivative test
A critical point of $f$ is a value $c$ where $f'(c) = 0$ or where $f'(c)$ does not exist. A local maximum or minimum can only happen at a critical point. The first derivative test looks at the sign of $f'$ on both sides. Plus to minus gives a local maximum, minus to plus gives a local minimum, and no change gives neither.
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 2about 25 min
The second derivative test and extreme values on an interval
The second derivative test: if $f'(c) = 0$ and $f''(c) < 0$ , then $c$ is a local maximum, and if $f''(c) > 0$ , a local minimum. If $f''(c) = 0$ the test says nothing, and the first derivative test is used. A continuous function on a closed interval $[a, b]$ always has a greatest and a least value. They are found among the end points and the critical points inside the interval.
1 simulation1 practice question1 Python program3 problems
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Lesson 3about 25 min
Optimisation problems
An optimisation problem asks for the largest or smallest value of a quantity, such as the volume of a box or the cost of a fence. Write the quantity as a function of one variable, using the condition that links the variables. Find the domain, locate the critical points, and check which one gives the maximum or minimum, using the second derivative or the end points. Then answer in the words and units of the question.
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
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Maxima, Minima and Optimisation: Chapter Test

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Maxima, Minima and Optimisation

Course outline (3 lessons)

Critical points and the first derivative test

The second derivative test and extreme values on an interval

Optimisation problems

Test Your Learning

Your teacher

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