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Critical number & maximum and minimum values
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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
critical number: a number c in the domain of a function f such that:

- IntroductionHow to describe graphs of functions?
a)∙ local maximum ∙ local minimum ∙ critical numberb)state the: ∙ absolute maximum ∙ absolute minimumc)on the interval,
−1≤x≤12state the: ∙ absolute maximum ∙ absolute minimum - 1.Find the critical numbers of the function:a)
f(x)=3x2−5xb)
f(x)=x31−x−32 - 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
- 3.f(x)=3x5−15x4+25x3−15x2+5a)Find the critical numbers.b)On what intervals is 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.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.
- 5.Find the absolute maximum and minimum values of the function:
f(x)=3x5−15x4+25x3−15x2+5−21≤x≤21
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36.
Derivative Applications
36.1
Rectilinear Motion: Derivative
36.2
Critical number & maximum and minimum values
36.3
Curve sketching