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Increasing / Decreasing Test

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Presentation on theme: "Increasing / Decreasing Test"— Presentation transcript:

1 Increasing / Decreasing Test
If f’(x) > 0 on an interval, then f is increasing on that interval. If f’(x) < 0 on an interval, then f is decreasing on that interval. ex: Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5 is increasing and decreasing… f ’(x) = 12x3 – 12x2 – 24x Factor: 12x(x – 2)(x + 1) Critical Numbers: x = -1, 0, 2 Test: ( - , -1) (-1, 0) (0, 2) (2, )

2 The First Derivative Test
For c being a critical number of a continuous function f… If f ’ changes from positive to negative at c, then f has a local maximum at c. If f ’ changes from negative to positive at c, then f has a local minimum at c. If f ’ does not change sign at c, then f has no local maximum or minimum at c. c2 c1 f ’ FROM – TO + = MIN FROM + TO – = MAX

3 ex: Critical Numbers: x = -1, 0, 2 Test: ( - , -1) (-1, 0) (0, 2) (2, ) -1 0 2 Concavity Test If f ’’(x) > 0 in [a,b], then f is concave up on [a,b] If f ’’(x) < 0 in [a,b], then f is concave down on [a,b]

4 The Second Derivative Test
For f ’’ being 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. f ’ f ’’ c1 c2 f

5 ex: For the curve f(x) = x4 - 4x3 use calculus to find local maxima and minima, concavity intervals and points of inflection… Critical #’s = 0, 3 [ Test f ’ ] (- , 0) f’(-1) = No Max / 0 f’(1) = (0, 3) (3, ) f’(4) = + NegPos = Local Min

6 + + Test second derivative for concavity intervals: [ Test f ’’ ]
2nd derivative test f ’ critical # 0 [ Test f ’’ ] (- , 0) f ’’(-1) = + f ’ critical # 3 = (36)(1) = 36, so f ’’ is + = 3 CC Inflection 0 f ’’(1) = (0, 2) CC Inflection 2 (2, ) f ’’(5) = + CC Summary Local min at (3, -27) No local max CC up from (– , 0) U (2, ) CC down from (0, 2)

7 f(x) = x4 - 4x3

8 nDeriv For f(x) = x2/3(6 – x)1/3 graph the function and its first and second derivatives…. Type f(x) into Y1 Go to Y2 line Go to MATH [8] (nDeriv) / Press ENTER Set nDeriv arguments to (Y1, X, X) *Y1 in VARS ~ Y-VARS ~ FUNCTION ~ #2 Repeat on Y3 line using (Y2, X, X) for nDeriv Graph


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