suppose f(x) =ax^blnx is a real- valued function. Determine exact values(not decimal approximations) fro nonzero constants a and b so that the function f has a critical point at x=e^3and a maximum value of 1/2e

If there is a critical value at x = e^{3} then f '(e^{3}) = 0. This allows you to find b. Also x = e^{3} is the only critical value of f(x) (Why?) and hence the maximum value you are given must occur ar x = e^{3}. Set f(e^{3}) = 1/2 e and solve for a.

Harley

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