Presentation on theme: "DO NOW: Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5"— Presentation transcript:
1DO NOW: Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5 is increasing and decreasing.
24.3 – Connecting f’ and f’’ with the Graph of f HW: Pg.
3First Derivative TestRecall that at a critical point, a function can have:A local maximum,A local minimum, orNeither.We can look at whether f’ changes sign at the critical point to decide which of the above possibilities is the case:
4First 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.
5Concavity In the figure on the next slide, the Slopes of the tangent lines increase from left to right on the interval (a,b), and soA 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.
6Concavity Concave ___________: F’ is decreasing f’’ < 0 F’ is increasing f’’ > 0
7Inflection PointA 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 TestIf 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.
8Determining 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.
9Example 1 Discuss y = x4 – 4x3 with respect to Concavity, Points of inflection, andLocal maxima and minima.