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

# Critical Numbers: The Key to Maximum and Minimum Values Unlock the power of critical numbers in calculus. Learn to identify key points, analyze function behavior, and solve real-world optimization problems with confidence and precision.

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Intros

Examples

- 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}$