How to find the exact location of all the relative and absolute extrema of the function?

Question

f(x) = x ln x with domain (0, +∞)?

Answer 1

The function is f(x) = x ln x  

The domain of the function is  (0 , +∞ ) .

To evaluate the relative and absolute extrema , first we have to find critical points .

f '(x) = x *(1/x) + ln x (1)          [ derivative of  uv = uv’ + vu’ ]        

f '(x) = 1 + ln x = 0 

1 + ln x = 0 

ln x = -1

x = e^(-1)

x = 0.3678 .

The function  f(x) = x ln x has one critical point that is x = 0.3678 .

Now put x = 0.3678 in f(x)  .

f(x) = 0.3678 ln (0.3678)

f(x) = 0.3678 (-1.000215)

f(x) = -0.367879 < 0

Hence f(x) has local (relative ) minimum at x = 0.3678 .

f(x) has a only one critical point , hence absolute minimum and relative minimum are same .

f(x) has absolute minimum and relative minimum at x = 0.3678 is f(x) = -0.367879 .

f(x) doesn't have absolute maximum and relative maximum .

Now graph the function f(x) = x ln x .

Clearly we can observe from the graph .

f(x) has absolute /relative minimum at x = 0.3678 is f(x) = -0.367879 .