Question

Write the point where f(x) = x log_ex attains minimum value.

Answer 1

Let f (x) = x loge x. 

Clearly,

f(x) is only defined for x > 0. 

For minimum value; 

f’(x) = 0 and f’’(x) > 0. 

f’(x) = 1 + ln x 

⇒ x = 1/e 

f’’(x) = 1/x, 

Clearly,

f’’(x) > 0 for all x. 

So, only minima is defined. 

So, minima of f (x) is at x = 1/e

f(\(\frac{1}{e}\)) = \(-\frac{1}{e}\)