4.3 Mean-Value Theorem and Monotonicity

Transcription

1 .3 Mean-Value Theorem and Monotonicit 1. Mean Value Theorem Theorem: Suppose that f is continuous on the interval a, b and differentiable on the interval a, b. Then there eists a number c in a, b such that f f b f a c. b a Observe that (1) Rolle s Theorem is a special case of the Mean Value Theorem. (2) The right side of the equation is the slope of the secant line to the curve f passing through points a, f a and b, f b. The left side of the equation is the slope of the tangent line to the curve f at the point c, f c

2 Algebraicall, we can find c in three steps: (i) Compute f. f b f a (ii) Compute d. b a (iii) Set f d, solve for, and c if it is in a, b. 2

3 Eample: Find all possible values of c satisfing the conclusion of Mean Value Theorem for f on the interval 0, 3. 2 f is a polnomial so it is continuous on 0, 3 2 0, 3. Find c : 2 (i) Compute f and is differentiable on (ii) Compute d f 3 2 f (iii) Set , solve for : , is outside 0, 3 and is in 0, 3. So, c

4 Eample: Find all possible values of c satisfing the conclusion of Mean Value Theorem for f sin on the interval 0, 2. (1) Compute f sin cos. (2) Compute d f 2 f sin (3) Find c in 0, 2 so that f c 1 sin cos 1, b Calculator

5 Eample: Eplain wh it is not valid to use the Mean-Value Theorem for f 1 3, 1, f 1 is not continuous at 3 0. Hence, the Mean-Value Theorem cannot be applied for this problem

6 2. Increasing and Decreasing Functions Definition: A function f is (strictl) increasing on an interval I if for ever 1, 2 in I with 1 2, f 1 f 2. A function f is (strictl) decreasing on an interval I if for ever 1, 2 in I with 1 2, f 2 f 1. Eample: The graph of f is given below f is increasing on 0,0.8, 2.5,. f is decreasing on 0.8,2.5,,. -10 f 6

7 Theorem: Suppose that f is differentiable on an interval I. (i) If f 0 for all in I, then f is increasing on I. (ii) If f 0 for all in I, then f is decreasing on I. Proof: Let 1 and 2 be in I and 1 2. Then b the Mean Value Theorem, we know there eists a value c in 1, 2 such that f c f 2 f If f c 0, then f 2 f that implies f 2 f 1 0 because 2 1. Hence, f 2 f 1, that is, f is increasing. In a similar wa, we can show (ii). Eample: The graph of f is given below. Determine graphicall the interval on which f is increasing. 7

8 Eample: The graph of f is given below. Determine graphicall the interval on which f is increasing f is increasing on 0,1.6, 3.1,.7 because f f 8

9 Eample: Find the intervals where f is increasing and decreasing. Verif answers with graphs of f and f. Step I: Compute f : f Step II: Find values of at which f 0: and 1. Step III: Check sign changes of f over intervals:,,,1, 1, f 5 36 f 0 2 f 2 36, interval,,1 1, sign of f So, f is increasing on,, 1, and is decreasing on, red f, green f 9

10 3. First Derivative Test For Critical Points: Theorem: Suppose that f is continuous on a, b and c in a, b is a critical point. (i) If f changes sign from positive to negative at c, then f c is a local maimum of f. The number c is called a local maimum point. (ii) If f changes sign from negative to positive at c, then f c is a local minimum of f The number c is called a local minimum point. (iii) If f does not change sign at c, then f c is not a local etremum. Eample: Let f 1 sin cos. Find all critical points and use the 1st Derivative Test to classif each as the location of a local maimum, local minimum of neither. 10

11 Eample: Let f 1 sin cos. Find all critical points and use the 1st Derivative Test to classif each as the location of a local maimum, local minimum of neither. Step I: Find the domain of f : D f,. Step II: Compute f and find all critical numbers: f cos sin. (1) Critical number of tpe (i): f 0 sin cos, or tan 1 or n, sa,..., 7, 3,, 5,... (2) Critical number of tpe (ii): None. Step III: Check sign change of f over intervals:..., 11, 7 7, 3, 3,,, 5, 5, 9,... f cos sin 1, f 0 cos 0 sin 0 1, f cos sin 1,... 11

12 interval 11, 7 7, 3 3,, 5 5, 9 sign of f 2n are local maimum points and 5 minimum points. 2n is a local 12

13 Eample: Let f 2 2 sketch the graph of f. Step I: horizontal and vertical asmptotes:. Find all asmptotes and etrema, and Horizontal asmptote: lim 2 2 1, 1 Vertical asmptote: 2 0, 2, 2 Step II: Compute f and find all critical numbers: f (1) Critical number of tpe (i): f 0, 8 0, 0. (2) Critical number of tpe (ii): f is not defined at 2but the are not in D f, so none. Step III: Check sign change of f over,0, 0, : f 0 for in, 0 and f 0 for in 0,. 0 is a local maimum point. 13

14 f, -.- f 1

15 Eample: Let f 5/3 3 2/3. Find all critical numbers and use the 1st Derivative Test to classif each as the location of a local maimum, local minimum of neither. Step I: Find the domain of f : D f,. Step II: Compute f and find all critical numbers: f 5 3 2/3 2 1/3 1/ (1) Critical number of tpe (i): f (2) Critical number of tpe (ii): f is not defined when 0. Step III: Check sign change of f over intervals:,0, 0, 6 5, 6 5, f f f , interval,0 0, 6 5 sign of f - 6 5, 15

16 So, 0 is a local maimum point and 6 5 point. is a local minimum f 16

17 Eample: Sketch a graph of a function with the given properties: (i) f 0 1, f 2 5 (ii) f 0 0, f 2 0 f 0, for 0 and 2; f 0 for 0 2 Eample: Sketch a graph of a function with the given properties: (i) f , f 3 0, f 0 does not eist (ii) f 3 0, f 0 does not eist f 0, for 0 and 3; f 0 for