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DO NOW:

DO NOW:. Find where the function f(x) = 3x 4 – 4x 3 – 12x 2 + 5 is increasing and decreasing. 4.3 – Connecting f ’ and f ’’ with the Graph of f. HW: Pg. First Derivative Test. Recall that at a critical point, a function can have: A local maximum, A local minimum, or Neither.

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  1. DO NOW: • Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5 is increasing and decreasing.

  2. 4.3 – Connecting f’ and f’’ with the Graph of f HW: Pg.

  3. First Derivative Test • Recall that at a critical point, a function can have: • A local maximum, • A local minimum, or • Neither. We can look at whether f’ changes sign at the critical point to decide which of the above possibilities is the case:

  4. First Derivative Test (cont’d) The First Derivative Test Suppose that c is a critical number of a continuous function f. • If f’ changes from positive to negative at c, then f has a local _____________ at c. • If f’ changes from negative to positive at c, then f has a local _____________ at c. • If f’ does not change sign at c (that is, f’ is positive on both sides of c or negative on both sides), then f has no local maximum or minimum at c.

  5. Concavity • In the figure on the next slide, the • Slopes of the tangent lines increase from left to right on the interval (a,b), and so A function (or its graph) is called concave upward on an interval I if i is an increasing function on I. It is called concave downward on I if f’ is decreasing on I.

  6. Concavity • Concave ___________: • F’ is decreasing  f’’ < 0 • Concave ___________: • F’ is increasing  f’’ > 0

  7. Inflection Point • A point where a curve changes its direction of concavity is called an inflection point. • Thus there is a point of inflection at any point where the second derivative changes sign. Concavity Test • If f’’(x) > 0 for all x in I, then the graph of f is concave upward on I. • If f’’(x) < 0 for all x in I, then the graph of f is concave downward on I.

  8. Determining Local Max/Min The Second Derivative Test Suppose f’’ is continuous near c. • If f’(c) = 0 and f’’(c) > 0, then f has a local minimum at c. • If f’(c) = 0 and f’’(c) < 0, then f has a local maximum at c.

  9. Example 1 • Discuss y = x4 – 4x3 with respect to • Concavity, • Points of inflection, and • Local maxima and minima.

  10. Example 1 (solution)

  11. Example 2 • Discuss f(x) = x2/3(6 – x)1/3 with respect to: • Concavity • Points of inflection, and • Local maxima and minima.

  12. Example 2 (solution)

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