Problem Set 11s - Hung Nguyen Problem Set 11 Solution 4.2...

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Hung Nguyen Problem Set 11 Solution GE 207C 4.2 Mean Value Theorem 4.3 How Derivatives Affect Graph 1 1. Find all numbers c that satisfy the conclusion of the Mean Value Theorem (MVT) on the given interval. a. 1 2 ) ( 2 + = x x f [0, 2] Solution: f ( x ) = 2 x 2 + 1 f '( x ) = 4 x f ( c ) = 4 c = f ( b ) f ( a ) b a = f (2) f (0) 2 = 9 1 2 = 4 = 4 c c = 1 b. x x x f + = 3 ) ( [1, 2] Solution: 53 . 1 3 7 3 7 8 1 2 10 1 3 1 ) 1 ( ) 2 ( 1 3 ) ( ' 1 3 ) ( ' ) ( 2 2 2 2 3 ± = ± = = = = + = + = + = + = c c c f f c c f x x f x x x f But 53 .. 1 is not within the interval so 53 . 1 = c c. 1 1 ) ( + = x x f [0,2] Solution: ( ) ( ) ( ) 3 1 3 1 3 1 3 1 2 1 3 / 1 1 1 ) ( ' 1 1 ) ( ' 1 1 ) ( 2 2 2 ± = ± = + = + = = + = + = + = c c c c c f x x f x x f Since 3 1 = c is not on the interval. 73 . 0 3 1 = + = c d. f ( x ) = x x + 2 [1, 4] Solution: 2. What on interval(s) is the function increasing? 3. Use the graph of the derivative of f to locate the critical points x 0 at which f has either a local maximum or a local minimum?
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Hung Nguyen Problem Set 11 Solution GE 207C 4.2 Mean Value Theorem 4.3 How Derivatives Affect Graph 2 4. If f (1) = 10 and f '( x ) 2 for 4 1
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