(a) Suppose f :R → R is a differentiable function of a single variable. Show that if f has a unique critical point at x0 that is the site of a strict local extremum of f , then f must attain a global extremum at x0.
(b) Let f (x, y) = 3yex − e3x − y3. Verify that f has a unique critical point and that f attains a local maximum there. However, show that f does not have a global maximum by considering how f behaves along the y-axis. Hence, the result of part (a) does not carry over to functions of more than one variable.
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