Critical number & maximum and minimum values - Derivative Applications

Critical number & maximum and minimum values

Another powerful usage of differential calculus is optimization, for example, finding the number of products needed to be sold at a store to maximize its monthly revenue or to minimize its monthly costs. In this section, we will link the application of differential calculus with finding the local extrema, the maxima and minima, of a function.

Lessons

Notes:

critical number: a number $\ c$ in the domain of a function $\ f$ such that:

Intro Lesson

How to describe graphs of functions?
1.
Find the critical numbers of the function:

3.
$f(x)=3x^{5}-15x^{4}+25x^{3}-15x^{2}+5$

Critical number & maximum and minimum values

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First Derivative Test
local maximum	local minimum	no maximum or minimum	no maximum or minimum

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Intro Lesson: a

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Intro Lesson: b

Intro Lesson: c

Lesson: 1a

Lesson: 1b

Lesson: 2

Lesson: 3a

Lesson: 3b

Lesson: 3c

Lesson: 3d

Lesson: 3e

Lesson: 4

Lesson: 5

Intro

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Don't just watch, practice makes perfect

Do better in math today

Don't just watch, practice makes perfect

Lessons

Notes:

critical number: a number c\ c c in the domain of a function f\ f f such that:

Intro Lesson

How to describe graphs of functions?

a)

∙\bullet ∙ local maximum ∙\bullet ∙ local minimum ∙\bullet ∙ critical number

b)

state the: ∙\bullet ∙ absolute maximum ∙\bullet ∙ absolute minimum

c)

on the interval,−1≤x≤12-1\leq x\leq 12\, −1≤x≤12state the: ∙\bullet ∙ absolute maximum ∙\bullet ∙ absolute minimum

1.

Find the critical numbers of the function:

a)

f(x)=3x2−5xf(x)={^3}\sqrt{x^{2}-5x}f(x)=3x2−5x​

b)

f(x)=x13−x−23f(x)=x^{\frac{1}{3}}-x^{-\frac{2}{3}}f(x)=x31​−x−32​

2.

First Derivative Test: a test to determine whether or not f\ f f has a local maximum or minimum at a critical number First Derivative Test local maximum local minimum no maximum or minimum no maximum or minimum

3.

f(x)=3x5−15x4+25x3−15x2+5f(x)=3x^{5}-15x^{4}+25x^{3}-15x^{2}+5 f(x)=3x5−15x4+25x3−15x2+5

a)

Find the critical numbers.

b)

On what intervals is f f f increasing or decreasing?

c)

Find the local maximum and minimum values.

d)

Sketch the graph.

e)

Find the absolute maximum and minimum values.

4.

5.

Find the absolute maximum and minimum values of the function: f(x)=3x5−15x4+25x3−15x2+5f(x)=3x^{5}-15x^{4}+25x^3-15x^2+5f(x)=3x5−15x4+25x3−15x2+5−12≤x≤12 -\frac{1}{2}\leq x\leq\frac{1}{2}−21​≤x≤21​

Critical number & maximum and minimum values

Don't just watch, practice makes perfect.

We have over 350 practice questions in Calculus for you to master.

or

critical number: a number $\ c$ in the domain of a function $\ f$ such that:

$\bullet$ local maximum
$\bullet$ local minimum
$\bullet$ critical number

state the:
$\bullet$ absolute maximum
$\bullet$ absolute minimum

on the interval,
$-1\leq x\leq 12\,$ state the:
$\bullet$ absolute maximum
$\bullet$ absolute minimum

$f(x)={^3}\sqrt{x^{2}-5x}$

$f(x)=x^{\frac{1}{3}}-x^{-\frac{2}{3}}$

First Derivative Test: a test to determine whether or not $\ f$ has a local maximum or minimum at a critical number

First Derivative Test

local maximum

local minimum

no maximum or minimum

no maximum or minimum

$f(x)=3x^{5}-15x^{4}+25x^{3}-15x^{2}+5$

On what intervals is $f$ increasing or decreasing?

Find the absolute maximum and minimum values of the function:

$f(x)=3x^{5}-15x^{4}+25x^3-15x^2+5$
$-\frac{1}{2}\leq x\leq\frac{1}{2}$