Applications of Differentiation

Calculus Chapter 3

Extrema on an Interval

Calculus 3.1

Calculus 3.1 3

Extrema

• Extreme values• Minimum (plural minima)• Maximum (plural maxima)

• Can be absolute (on a closed interval) or relative (on an open interval)

Calculus 3.1 4

Critical numbers

• Where the derivative is• zero• Or undefined

• Relative extrema occur only at critical numbers

Calculus 3.1 5

Finding extrema on a closed interval

1. Find the critical numbers2. Evaluate the function at the critical

numbers3. Evaluate the function at the endpoints

of the interval4. The least of these values is the

minimum. The greatest is the maximum

Calculus 3.1 6

Examples

• Find the absolute extrema of the functions on the given intervals

2 5, 0,5

3

xf x

3 12 , 0,4f x x x

2 2 4 , 1,1f x x x