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Published byNoel nolan Easton Modified over 2 years ago

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Maximum ??? Minimum??? How can we tell? and decreasing just to the right of c, then f has a local minimum at c If f is increasing just to the left of a critical number c and increasing just to the right of c, then f has a local maximum at c If f is decreasing just to the left of a critical number c

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First Derivative Test Let c be a critical number of a continuous function f. 1.If f ' (x) changes from positive to negative at c, then f has a local maximum at c.

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First Derivative Test Let c be a critical number of a continuous function f. 2. If f ' (x) changes from negative to positive at c, then f has a local minimum at c.

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First Derivative Test Let c be a critical number of a continuous function f. 3. If f ' (x) does not change sign at c, then f has no maximum or minimum at c.

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EXAMPLE 1: f(x) = 3x 2 – 4x + 13 f ′(x) = 6x – 4 6x – 4 = 0 (critical number) f ′(x) < 0 local minimum at 6x – 4 < 0 6x < 4 6x – 4 > 0 6x > 4 tangent slope is positive tangent slope is negative f ′(x) > 0

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EXAMPLE 2: f(x) = x 3 – 12x – 5 f ′(x) = 3x 2 – 12 3x 2 – 12 = 0 3(x 2 – 4) = 0 3(x – 2)(x + 2) = 0 x = 2 or x = –2 – Test for x < –2 3( – 2)( + 2) Test for –2 < x < 2 3( – 2)( + 2) Test for x > 2 3( – 2)( + 2) max min Local minimum value is -21 f (–2) 11 f (2) = -21 Local maximum at value is 11 Critical values

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EXAMPLE 3 f(x) = x 4 – x 3 f ′(x) = 4x 3 – 3x 2 4x 3 – 3x 2 = 0 x 2 (4x – 3) = 0 critical values x = 0 or x = ¾ f (0) = 0f ( ¾ ) = 0.11 Test for x < 0 ( ) 2 (4( ) – 3) Test for 0 < x < ¾ ( ) 2 (4( ) – 3) Test for x > ¾ ( ) 2 (4( ) – 3) 0 + Local minimum value is 0.11 Local maximum DNE

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