Math CentralQuandaries & Queries


Question from charles, a student:

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

Hi Charles,

If f(x) = axb ln(x) then

f '(x) = abxb-1 ln(x) + axb-1 = axb-1 (b ln(x) + 1).

If there is a critical value at x = e3 then f '(e3) = 0. This allows you to find b. Also x = e3 is the only critical value of f(x) (Why?) and hence the maximum value you are given must occur ar x = e3. Set f(e3) = 1/2 e and solve for a.


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