Midterm 2 Solutions - Test 2 Solutions Math 102 Problem 1 5...

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Unformatted text preview: Test 2 - Solutions Math 102 Problem 1. ( 5 points ) Find the exact interval on which f ( x ) = x 4- 6 x 2 + 5 is concave down. A:- 5 < x < 1 5 , B:- 4 < x < 1 4 , C:- 3 < x < 1 3 , D:- 2 < x < 1 2 , E:- 1 < x < 1 F:- 1 2 < x < 2 , G:- 1 3 < x < 3 , H:- 1 4 < x < 4 , I:- 1 5 < x < 5 , J:- ∞ < x < ∞ Solution. We need to find the interval where f 00 ( x ) is negative. We have f ( x ) = 4 x 3- 12 x , and f 00 ( x ) = 12 x 2- 12. So the answer is E . Remark: A very easy problem. 1 Problem 2. ( 5 points ) Consider the function g ( x ) = 2 x 2 ( 1- x 2 ) a) Find the critical numbers of g ( x ) . b) Using the 2nd derivative test, determine which ones yield relative minima, and which ones yield relative maxima for g ( x ) . Solution. a) From g ( x ) = 2 ( x 2- x 4 ) , we get g ( x ) = 2 ( 2 x- 4 x 3 ) = 4 ( x- 2 x 3 ) . The critical numbers come from solving g ( x ) = 0, i.e., x = 0, x = ± 1 / √ 2. b) As g 00 ( x ) = 4 ( 1- 6 x 2 ) , we get: • g 00 ( ) = 4 > 0, so 0 yields a relative minimum for g ( x ) • g 00 ( ± 1 / √ 2 ) =- 8 < 0, so both 1 / √ 2 and- 1 / √ 2 yield relative maxima for g ( x ) . Remark: An easy problem. 2 Problem 3. ( 5 points ) A new competitor on the energy-drink market wants a container in the shape of a right circular cylinder for their 666ml drink. What dimensions would you suggest for the can?a right circular cylinder for their 666ml drink....
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Midterm 2 Solutions - Test 2 Solutions Math 102 Problem 1 5...

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