DO NOW: Find where the function f(x) = 3x4 – 4x3 – 12x2 + 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. We can look at whether f’ changes sign at the critical point to decide which of the above possibilities is the case:
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.
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.
Concavity Concave ___________: F’ is decreasing f’’ < 0 F’ is increasing f’’ > 0
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.
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.
Example 1 Discuss y = x4 – 4x3 with respect to Concavity, Points of inflection, and Local maxima and minima.
Example 1 (solution)
Example 2 Discuss f(x) = x2/3(6 – x)1/3 with respect to: Concavity Points of inflection, and Local maxima and minima.
Example 2 (solution)