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DO NOW: Find where the function f(x) = 3x 4 – 4x 3 – 12x is increasing and decreasing.

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

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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:

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First Derivative Test (cont’d) The First Derivative Test Suppose that c is a critical number of a continuous function f. (a) If f’ changes from positive to negative at c, then f has a local _____________ at c. (b) If f’ changes from negative to positive at c, then f has a local _____________ at c. (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.

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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.

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Concavity Concave ___________: F’ is decreasing f’’ < 0 Concave ___________: F’ is increasing f’’ > 0

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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 (a) If f’’(x) > 0 for all x in I, then the graph of f is concave upward on I. (b) If f’’(x) < 0 for all x in I, then the graph of f is concave downward on I.

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

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Example 1 Discuss y = x 4 – 4x 3 with respect to Concavity, Points of inflection, and Local maxima and minima.

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Example 1 (solution)

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

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Example 2 (solution)

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