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Question

Write the maximum value of f(x) = x1/x.

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Solution

Given: fx = x1xTaking log on both sides, we getlog fx=1xlog xDifferentiating w.r.t. x, we get1fxf'x=-1x2log x+1x2f'x=fx1x21-log xf'x=x1x1x2-1x2log x ...1f'x=x1x-21-log x For a local maxima or a local minima, we must havef'x=0x1x-21-log x=0log x=1x=eNow, f''x =x1x1x2-1x2log x2+x1x-2x3+2x3log x-1x3=x1x1x2-1x2log x2+x1x-3x3+2x3log xAt x=ef''e =e1e1e2-1e2log e2+e1e-3e3+2e3log e=-e1e1e3<0So, x=e is a point of local maximum.Thus, the maximum value is given byfe = e1e

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