Relative Extrema Lesson 5.2.

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Presentation transcript:

Relative Extrema Lesson 5.2

Video Profits Revisited Recall our Digitari manufacturer Cost and revenue functions C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250 R(x) = 8.4x - .002x2 0 ≤ x ≤ 2250 Cost, revenue, and profit functions

Video Profits Revisited Digitari wants to know how many to make and sell for maximum profit Maximum profit when Profits neither increasing nor decreasing Slope = 0 Profits increasing on this interval Slope > 0 Profits decreasing on this interval Slope < 0

Relative Maximum Given f(x) on open interval (a, b) with point c in the interval Then f(c) is the relative max if f(x) ≤ f(c) for all x in (a, b) ( ) a c b

Relative Minimum Given f(x) on open interval (a, b) with point c in the interval Then f(c) is the relative min if f(x) ≥ f(c) for all x in (a, b) c b ( ) a

Relative Max, Min Note Important Rule: Relative max or min does not guarantee f '(x) = 0 Important Rule: If a function has a relative extremum at c Then either c a critical number or c is an endpoint of the domain

First Derivative Test Given f(c) is relative max if f(x) differentiable on (a, b), except possibly at c c is only critical number in interval f(c) is relative max if f '(x) > 0 on (a, c) and f '(x) < 0 on (c, b) ( ) a c b

First Derivative Test Given f(c) is relative min if f(x) differentiable on (a, b), except possibly at c c is only critical number in interval f(c) is relative min if f '(x) < 0 on (a, c) and f '(x) > 0 on (c, b) c b ( ) a

First Derivative Test Note two other possibilities f '(x) < 0 on both sides of critical point f '(x) > 0 on both sides of critical point Then no relative extrema

Finding Relative Extrema Strategy Find critical points Check f '(x) on either side Negative on left, positive on right → min Positive on left, negative on right → max Try it!

Application Back to Digitari … cost and revenue functions C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250 R(x) = 8.4x - .002x2 0 ≤ x ≤ 2250 Just what is that number of units to market for maximum profit? What is the maximum profit?

Assignment Lesson 5.2 Page 327 Exercises 1 – 53 EOO