Critical Numbers – Relative Maximum and Minimum Points Section 2.5 Critical Numbers – Relative Maximum and Minimum Points Note: This is two day lecture, marked by 2.5.1 and 2.5.2

If f ‘ (a) > 0 The graph of f(x) is INCREASING at x = a x = a

If f ‘ (a) < 0 The graph of f(x) is DECREASING at x = a x = a

If f ‘ (a) = 0, a maximum or minimum MAY exist. This is the graph of g(x) This is the graph of f(x)

If f ‘ (a) = 0, a maximum or minimum MAY exist. This is the graph of f ‘ (x) This is the graph of g ‘ (x)

A change from increasing to decreasing indicates a maximum

A change from decreasing to increasing indicates a minimum

FACTS ABOUT f ‘ (x) = 0 If f ‘ (x) > 0 on an interval (a, b), f is increasing on that interval If f ‘ (x) < 0 on an interval (a, b), f is decreasing on that interval If f ‘ (c) = 0 or f ‘ (c) does not exist, c is a critical number If f ‘ (c) = 0, a relative max or min will exist IF f ‘ (x) changes from positive to negative OR negative to positive. A RELATIVE max/min is a high/low point around the area. An ABSOLUTE max/min is THE high/low point on an interval.

This is the graph of f(x) on the interval [-1, 5]. Where are the relative extrema of f(x)? x = -1, x = 1, x = 3, x = 5 For what value(s) of x is f ‘ (x) < 0? (1, 3) For what value(s) of x is f ‘ (x) > 0? (-1, 1) and (3, 5) D. Where are the zero(s) of f(x)? x = 0 This is the graph of f(x) on the interval [-1, 5].

This is the graph of f ‘ (x) on the interval [-1, 5]. (-1, 1), (3, 5) Where are the relative extrema of f(x)? x = -1, x = 0, x = 5 B. For what values of x is f ‘ (x) < 0? [-1, 0) C. For what values of x is f ‘ (x) > 0? (0, 5] For what values of x is f “ (x) > 0? This is the graph of f ‘ (x) on the interval [-1, 5]. (-1, 1), (3, 5)

This is the graph of f(x) on [-10, 3]. Where are the relative extrema of f(x)? x = -10, x = 3 On what interval(s) of x is f ‘ (x) constant? (-10, 0) On what interval(s) is f ‘ (x) > 0? For what value(s) of x is f ‘ (x) undefined? x = -10, x = 0, x = 3 This is the graph of f(x) on [-10, 3].

This is the graph of f ‘ (x) on [-10, 3]. Where are the relative extrema of f(x)? x = -10, x = -1, x = 3 On what interval(s) of x is f ‘ (x) constant? none On what interval(s) is f ‘ (x) > 0? For what value(s) of x is f ‘ (x) undefined? none This is the graph of f ‘ (x) on [-10, 3].

CALCULATOR REQUIRED Based upon the graph of f ‘ (x) given on the interval [0, 2pi], answer the following: Where does f achieve a minimum value? Round your answer to three decimal places. 3.665, 6.283 Where does f achieve a maximum value? Round your answer to three decimal places. 0, 5.760

Given the graph of f(x) on to the right, answer the two questions below. Estimate to one decimal place the critical numbers of f(x). -1.4, -0.4, 0.4, 1.6 Estimate to one decimal place the value(s) of x at which there is a relative maximum. -1.4, 0.4

Given the graph of f ‘ (x) on to the right, answer the three questions below. Estimate to one decimal place the critical numbers of f(x). -1.9, 1.1, 1.8 Estimate to one decimal place the value(s) of x at which there is a relative maximum. 1.1 Estimate to one decimal place the value(s) of x at which there is a relative minimum. -1.9, 1.8

CALCULATOR REQUIRED a) For what value(s) of x will there be a horizontal tangent? 1 b) For what value(s) of x will the graph be increasing? c) For what value(s) of x will there be a relative minimum? 1 d) For what value(s) of x will there be a relative maximum? none

This is the graph of f(x) on For what value(s) of x is f ‘ (x) = 0?   On what interval(s) is f(x) increasing? . Where are the relative maxima of f(x)? -1 and 2 -1, 4 (-3, -1), (2, 4) This is the graph of f(x) on [-3, 4].

This is the graph of f ‘ (x) For what value(s) of x if f ‘ (x) = 0?   For what value(s) of x does a relative maximum of f(x) exist? For what value(s) of x is the graph of f(x) increasing? For what value(s) of x is the graph of f(x) concave up? -2, 1 and 3 -3, 1, 4 (-2, 1), (3, 4] This is the graph of f ‘ (x) [-3, 4] (-3, -1) U (2, 4)

This is the graph of f(x) on For what values of x if f ‘(x) undefined?   For what values of x is f(x) increasing? For what values of x is f ‘ (x) < 0? Find the maximum value of f(x). 6 (-5, 1) (1, 3) -5, 1, 3 This is the graph of f(x) on [-5, 3]

This is the graph of f ‘ (x) For what value(s) of x is f ‘ (x) undefined?   For what values of x is f ‘ (x) > 0? On what interval(s) is the graph of f(x) decreasing? On what interval(s) is the graph of f(x) concave up? (0, 7) (-7, 0) (0, 7] none This is the graph of f ‘ (x) on [-7, 7].

This is the graph of f(x) on For what value(s) of x is f ‘ (x) = 0?   For what value(s) of x does a relative minimum exist? On what intervals is f ‘ (x) > 0? f “ (x) > 0? (-1, 1), (1, 2) (-2, -1.5), (-0.5, 0.5), (1.5, 2) -2, -0.5, 1.5 -1.5, -0.5, 0.5, 1.5 This is the graph of f(x) on [-2, 2].

This is the graph of f ‘ (x) on For what value(s) of x is f ‘ (x) = 0?   For what value(s) of x is there a local minimum? f ‘ (x) > 0? f “ (x) > 0? (-2, -1.5), (-0.5, 0.5), (1.5, 2) (-2, -1), (0, 1) -2, 0, 2 -2, -1, 0, 1, 2 This is the graph of f ‘ (x) on [-2, 2]