# Critical number & maximum and minimum values

## Everything You Need in One PlaceHomework problems? Exam preparation? Trying to grasp a concept or just brushing up the basics? Our extensive help & practice library have got you covered. | ## Learn and Practice With EaseOur proven video lessons ease you through problems quickly, and you get tonnes of friendly practice on questions that trip students up on tests and finals. | ## Instant and Unlimited HelpOur personalized learning platform enables you to instantly find the exact walkthrough to your specific type of question. Activate unlimited help now! |

#### Make math click 🤔 and get better grades! 💯Join for Free

Get the most by viewing this topic in your current grade. __Pick your course now__.

##### Intros

##### Examples

###### Lessons

- Find the critical numbers of the function:
- 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

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