Critical number & maximum and minimum values
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- 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)=3x5−15x4+25x3−15x2+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)=3x5−15x4+25x3−15x2+5−21≤x≤21
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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:
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