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# handout_10_30 - g x = 2[Hint Think of what would happen if...

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Math 1a: The mean value theorem and shapes of curves October 30, 2009 1. Consider the function f ( x ) = x 3 - 5 x 2 + 3 x + 1. Where is f increasing/decreasing? Where is f concave up/concave down? What are the local maxima and minima? What are the inflection points? 2. Suppose we know that f is a differentiable function such that f (0) = 0 and f 0 ( x ) < 3 for all x . What is the maximum possible value for f (5)? 3. Carry out the analysis as in Problem 1 for the function f ( x ) = xe x . 1

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4. Suppose that g ( x ) is differentiable, g 0 ( x ) is continuous and g 0 ( x ) 6 = 0 for all x . What is the maximum number of solutions of the equation
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Unformatted text preview: g ( x ) = 2? [Hint: Think of what would happen if g ( x ) = 2 for more than one value of x .] 5. Carry out the analysis as in Problem 1 for the function f ( x ) = x + √ 1-x , deﬁned for 0 ≤ x ≤ 1. Where does f attain its global maxima/minima? 6. Find the critical values of the function f ( x ) = x 4 ( x-1) 3 . What does the second derivative test say about their nature (i.e. are they local maxima/minima/neither/can’t say)? 2...
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handout_10_30 - g x = 2[Hint Think of what would happen if...

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