# Critical number & maximum and minimum values

0/3
##### Introduction
###### Lessons
1. How to describe graphs of functions? 2. $\bullet$ local maximum
$\bullet$ local minimum
$\bullet$ critical number

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

4. on the interval,
$-1\leq x\leq 12\,$state the:
$\bullet$ absolute maximum
$\bullet$ absolute minimum
0/10
##### Examples
###### Lessons
1. Find the critical numbers of the function:

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

2. $f(x)=x^{\frac{1}{3}}-x^{-\frac{2}{3}}$
2. 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
1. $f(x)=3x^{5}-15x^{4}+25x^{3}-15x^{2}+5$
1. Find the critical numbers.
2. On what intervals is $f$ increasing or decreasing?
3. Find the local maximum and minimum values.
4. Sketch the graph.
5. Find the absolute maximum and minimum values.
2. The Closed Interval Method
To find the absolute maximum and minimum values of a continuous function $f$ on a closed interval [a, b]:
1.Find the values of $f$ at the critical numbers of $f$ in (a, b).
2.Find the values of $f$ at the left-endpoint and right-endpoint of the interval
3.Compare all values from steps 1 and 2: the largest$\$is the absolute maximum value; the smallest$\$is the absolute minimum value.
1. Find the absolute maximum and minimum values of the function:

$f(x)=3x^{5}-15x^{4}+25x^3-15x^2+5$
$-\frac{1}{2}\leq x\leq\frac{1}{2}$
###### Free to Join!
StudyPug is a learning help platform covering math and science from grade 4 all the way to second year university. Our video tutorials, unlimited practice problems, and step-by-step explanations provide you or your child with all the help you need to master concepts. On top of that, it's fun - with achievements, customizable avatars, and awards to keep you motivated.
• #### Easily See Your Progress We track the progress you've made on a topic so you know what you've done. From the course view you can easily see what topics have what and the progress you've made on them. Fill the rings to completely master that section or mouse over the icon to see more details.
• #### Make Use of Our Learning Aids   ###### Practice Accuracy

See how well your practice sessions are going over time.

Stay on track with our daily recommendations.

• #### Earn Achievements as You Learn   Make the most of your time as you use StudyPug to help you achieve your goals. Earn fun little badges the more you watch, practice, and use our service.
• #### Create and Customize Your Avatar   Play with our fun little avatar builder to create and customize your own avatar on StudyPug. Choose your face, eye colour, hair colour and style, and background. Unlock more options the more you use StudyPug.
###### Topic Notes
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.

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