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Relative Extrema

Relative Extrema. Lesson 5.5. Video Profits Revisited. Recall our Digitari manufacturer Cost and revenue functions C(x) = 4.8x - .0004x 2 0 ≤ x ≤ 2250 R(x) = 8.4x - .002x 2 0 ≤ x ≤ 2250 Cost, revenue, and profit functions. Maximum profit when

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Relative Extrema

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  1. Relative Extrema Lesson 5.5

  2. Video Profits Revisited • Recall our Digitari manufacturer • Cost and revenue functions • C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250 • R(x) = 8.4x - .002x20 ≤ x ≤ 2250 • Cost, revenue, and profit functions

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

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

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

  6. Relative Max, Min • Note • 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

  7. First Derivative Test Given • 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) ( ) c b a

  8. First Derivative Test Given • 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

  9. First Derivative Test • Note two other possibilities • f '(x) < 0 on both sidesof critical point • f '(x) > 0 on both sidesof critical point • Then no relative extrema

  10. 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!

  11. Application • Back to Digitari … cost and revenue functions • C(x) = 4.8x - .0004x2 0 ≤ x ≤ 2250 • R(x) = 8.4x - .002x20 ≤ x ≤ 2250 • Just what is that number of units to market for maximum profit? • What is the maximum profit?

  12. Assignment • Lesson 5.2 • Page 327 • Exercises 1 – 53 EOO

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