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# 12.05 - For Wednesday do true/false questions for chapter 4...

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For Wednesday: do true/false questions for chapter 4. Also, does there exist a twice-differentiable function f on R satisfying f ( x ) < 0 and f ′′ ( x ) > 0 for all x ? Section 4.7: Antiderivatives (concluded) Last time we asked “Does the function f ( x ) = | x | have an antiderivative on R ?” and found that the answer was Yes; the general form was … ..?. . F ( x ) = x | x | / 2 + C . Does the Heaviside function {0 if x < 0 f ( x ) = { {1 if x 0 have an antiderivative on R ? . .. ..?. . No. Why not? ..?. . Is it enough to say “The Heaviside function is discontinuous, so it doesn’t have an antiderivative on the whole real line R ?” . .. No, because some discontinuous functions have

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antiderivatives (see the discussion of the badly behaved function 2 x sin (1/ x ) – cos (1/ x ) earlier); that is, a discontinuous function can still be the derivative of some differentiable function. Suppose there were an antiderivative of
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12.05 - For Wednesday do true/false questions for chapter 4...

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