MATH 12002 - CALCULUS I 3.3: Increasing & Decreasing Functions - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §3.3: Increasing & Decreasing Functions

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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First Derivative Tests

Increasing/Decreasing Test

Let y = f (x) be a function. If f ′(x) > 0 on an interval I, then f is increasing on I. If f ′(x) < 0 on an interval I, then f is decreasing on I.

First Derivative Test for Local Max/Min

Let c be a critical number of a continuous function f . If f ′ changes sign from positive to negative at x = c, then f has a local maximum at c. If f ′ changes sign from negative to positive at x = c, then f has a local minimum at c. If f ′ does not change sign at x = c, then f has neither a local maximum nor a local minimum at c.

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Example

Example

Let f (x) = 3x4 − 20x3 + 36x2 − 17. Determine intervals where f is increasing, intervals where f is decreasing, and the location of all local maxima and minima. We need to determine the signs of the derivative of f . We have f ′(x) = 12x3 − 60x2 + 72x = 12x(x2 − 5x + 6) = 12x(x − 2)(x − 3). The critical numbers of f are therefore x = 0, x = 2, and x = 3. Since f ′ is a polynomial, it is continuous everywhere, and so can change sign only where it is 0; that is, at x = 0, 2, or 3. We will use a sign chart to determine the signs of f ′.

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Example

f (x) = 3x4 − 20x3 + 36x2 − 17 f ′(x) = 12x(x − 2)(x − 3)

2 3 12x x − 2 x − 3 f ′(x)

Inc-Dec Shape

− + + + − − + + − − − + − + − + D I D I ❅ ❅

• ❅

❅

• MIN

MAX MIN

f is increasing on (0, 2) ∪ (3, ∞); f is decreasing on (−∞, 0) ∪ (2, 3). Local maximum at x = 2; local minimum at x = 0 and at x = 3.

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Examples

In order to sketch the graph of f , we will need to plot the points whose x coordinates are in the sign chart. We need to evaluate f (x) = 3x4 − 20x3 + 36x2 − 17 at these points: f (0) = 3(04) − 20(03) + 36(02) − 17 = −17 f (2) = 3(24) − 20(23) + 36(22) − 17 = 48 − 160 + 144 − 17 = 15 f (3) = 3(34) − 20(33) + 36(32) − 17 = 243 − 540 + 324 − 17 = 10 Hence the points (0, −17), (2, 15), and (3, 10) are on the graph.

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Examples

2 3

Inc-Dec Shape

D I D I

MIN MAX MIN

❅ ❅

• ❅

❅

• ✲

✛ ✻ ❄

−20 −15 −10 −5 5 10 15 20 −1 1 2 3 4

q

(0, −17), MIN

q

(2, 15), MAX

q

(3, 10), MIN D.L. White (Kent State University) 6 / 11

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