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Review 3.1-3.3 - Increasing or Decreasing - Relative Extrema - Absolute Extrema - Concavity

Review 3.1-3.3 - Increasing or Decreasing - Relative Extrema - Absolute Extrema - Concavity. Increasing/Decreasing/Constant. Increasing/Decreasing/Constant. Increasing/Decreasing/Constant. Generic Example. The corresponding values of x are called Critical Points of f.

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Review 3.1-3.3 - Increasing or Decreasing - Relative Extrema - Absolute Extrema - Concavity

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  1. Review 3.1-3.3- Increasing or Decreasing- Relative Extrema- Absolute Extrema- Concavity

  2. Increasing/Decreasing/Constant

  3. Increasing/Decreasing/Constant

  4. Increasing/Decreasing/Constant

  5. Generic Example The corresponding values of x are called Critical Points of f

  6. Critical Points of f A critical number of a function f is a number cin the domain off such that

  7. Example Find all the critical numbers of When set = 0 Excluded values

  8. Determine where the function is increasing or decreasing: Plug into the derivate around each critical # - - - - - + + + + + + -1 0 Increasing: (-1, ∞) Decreasing: (-∞, -1)

  9. Example Find all the critical numbers of When set = 0 Excluded values

  10. Determine where the function is increasing or decreasing: Plug into the derivate around each critical # - - - - - + + + Not in Domain Not in Domain -1 0 1 Increasing: (-1, 0) Decreasing: (0, 1)

  11. Example Find all the critical numbers of When set = 0 Excluded values

  12. Determine where the function is increasing or decreasing: Plug into the derivate around each critical # - - - - - + + + + + + - - - - - + + + + + + -1 0 1 Increasing: (- ∞, -1) U (1, ∞) Decreasing: (-1, 1)

  13. Local min. Local max. Graph of

  14. If the price of a certain item is p(x) and the total cost to produce x units is C(x), at what production levels is profit increasing and decreasing? Now find P’(x) Now test around 18, -2 Increasing: (0, 18) Decreasing: (18, ∞)

  15. Relative Extrema A function f has a relative (local) maximum at x=c if there exists an open interval (r, s) containing c suchthat f (x)= f (c) Relative Maxima

  16. Relative Extrema A function f has a relative (local) minimum at x=c if there exists an open interval (r, s) containing c suchthat f (c) = f (x) Relative Minima

  17. The First Derivative Test Determine the sign of the derivative of f to the left and right of the critical point. left right conclusion f (c) is a relative maximum f (c) is a relative minimum No change No relative extremum

  18. Relative max. f (0) = 1 Relative min. f (4) = -31 The First Derivative Test Find all the relative extrema of Excluded Values: None + 0 - 0 + 0 4

  19. The First Derivative Test

  20. The First Derivative Test Find all the relative extrema of Excluded Values: None - - - - - + + + - - - - - + + + -1 0 1 Rel. Min. (1, -2) Rel. Max. (-1, 2)

  21. Example from before: Relative max. Relative min. Exclude Values: + ND + 0 - ND - 0 + ND + -1 0 1

  22. Rel. min. Rel. max. Graph of

  23. Absolute Extrema Let f be a functiondefined on a domain D Absolute Maximum Absolute Minimum

  24. Absolute Extrema A function f has an absolute (global) maximum atx = c if f (x)= f (c)for allx in the domain D of f. The number f (c) is called the absolute maximumvalue of f in D Absolute Maximum

  25. Absolute Extrema A function f has an absolute (global) minimum atx = c if f (c)= f (x)for allx in the domain D of f. The number f (c) is called the absolute minimumvalue of f in D Absolute Minimum

  26. Finding absolute extrema on [a,b] • Find all critical numbers for f (x) in (a,b). • Evaluate f (x) for all critical numbers in (a,b). • Evaluate f (x) for the endpoints a and b of the interval [a,b]. • The largest value found in steps 2 and 3 is the absolute maximum for f on the interval [a , b], and the smallest value found is the absolute minimum for f on [a,b].

  27. Absolute Max. Absolute Min. Absolute Max. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Evaluate

  28. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,3) are x = 0, 2 Absolute Max. Absolute Min.

  29. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Max. Evaluate Absolute Min.

  30. Example Find the absolute extrema of Critical values of f inside the interval (-1/2,1) is x = 0 only Absolute Max. Absolute Min.

  31. Concavity Let f be a differentiable function on (a, b). 1.f is concave upward on (a, b) if f' is increasing on aa(a, b). That is f ''(x)>0 for each value of x in (a, b). 2.f is concave downward on (a, b) if f' is decreasing on (a, b). That is f ''(x)< 0 for each value of x in (a, b). concave upward concave downward

  32. Inflection Point A point on the graph of f at which fis continuousandconcavity changes is called an inflection point. To search for inflection points, find any point, c in the domain where f ''(x)=0 or f ''(x)is undefined. If f ''changes sign from the left to the right of c, then (c,f (c))is an inflection point of f.

  33. Example: Inflection Points Find all inflection points of

  34. Inflection point at x 2 - 0 + 2

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