Sometimes, odd behavior can be hidden beneath the surface of a rather normal-looking
Chapter 2, Problem 16(choose chapter or problem)
Sometimes, odd behavior can be hidden beneath the surface of a rather normal-looking function. Consider the following function: f(x) = 0 if x < 0 x2 if x 0. (a) Sketch a graph of this function. Does it have any vertical segments or corners? Is it differentiable everywhere? If so, sketch the derivative f of this function. [Hint: You may want to use the result of Example 4 on page 94.] (b) Is the derivative function, f (x), differentiable everywhere? If not, at what point(s) is it not differentiable? Draw the second derivative of f(x) wherever it exists. Is the second derivative function, f(x), differentiable? Continuous?
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