Question

Discuss extreme values of the function f(x) = x.logx

Answer 1

f(x) = x.logx

Differentiating w.r.t. x, 

`f'(x) = x . 1/x + logx.1`

f'(x) = 1 + logx

Differentiating again w.r.t. x, 

`f''(x) = 1/x`

For maxima or minima, 

f'(x) = 0

∴ 1 + logx = 0

∴ logx = -1

∴ x = `e^-1`

∴ `f''(1/e) = 1/(1/e)`

∴ `f''(1/e) = e`

∴ `f''(1/e) > 0`

∴ f(x) is minimum at x = `1/e`