(Solution) - Suppose f x is a positive function with a maximum at -(2025 Original AI-Free Solution)

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Suppose f(x) is a positive function with a maximum at x*. We can often find maxima and minima of other functions composed with f(x). For each of the functions h(x) = g(f(x)),
a. Show that h has a critical point at x*.
b. Compute the second derivative at this point.
c. Check whether your function has a minimum or a maximum and explain.
d. Check your result using the function f(x) = xe-x for x ? 0 which has a maximum at x = 1. Sketch a graph of f(x) and h(x) in this case.
19. g(f) = 1/f.
20. g(f) = 1 - f.
21. g(f) = ln(f).
22. g(f) = f - f2.