Show that the function given by has maximum at x = e.

To find where f(x) is maximum we must first find the critical points where f’(x) = 0

Differentiate with respect to x and equate to 0

Using rule which says that

Here u = logx and v = x

⇒ logx = 1

⇒ x = e

So we have only one point to check whether its point of maxima or minima and that will be the maximum/minimum of f(x)

Now x = e is a point of maxima iff f’’(x) is negative at x = e(that is f’’(e) < 0)

We have calculated earlier f’(x)

Differentiate again using the rule

As we have to check f’’(e) put x = e

Hence f’’(e) < 0 and hence x = e is a point of maxima

Hence proved