Warm up Problems Let f (x) = x 3 – 3x + 1. 1) Find and classify all critical points. 2) Find all inflection points.

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Presentation transcript:

Warm up Problems Let f (x) = x 3 – 3x ) Find and classify all critical points. 2) Find all inflection points.

Graph of a Function, Part 2

Second Derivative Test If p is a critical point of f (x) and f  (p) < 0, then p is a local maximum. If p is a critical point of f (x) and f  (p) > 0, then p is a local minimum.

Ex. Find and classify all critical points of f (x) = x 3 – 5x 2 + 3x – 1.

Def. The absolute maximum (global max) value of a function on an interval is the largest value that the function attains. Def. The absolute minimum (global min) value of a function on an interval is the smallest value that the function attains.

Thm. The absolute max. and min. will occur at one of the following: the point p where f (p) = 0 the point p where f (p) is undef. an endpoint of the interval critical points

Ex. Find the absolute max. and min. values of f (x) = 3x 5 – 5x 3 – 1 on.

Ex. Find the x-coordinate of all local max./min. and absolute max./min. of f (x) = x 2 for -2 ≤ x ≤ 0 by graphing.  What about open intervals?

Ex. Find the x-coordinate of the absolute minimum of f (x) on [0,5]. Justify your answer.

Ex. Find the x-coordinate of the absolute maximum of g(x). Justify your answer.

Ex. For a particle moving along the x-axis, you are given the graph of the velocity below. Assume x(1) = 10. a)Find the total distance travelled on [1,7]. b)Find x(7). c)When is the particle farthest to the left on [1,7]?