Question

What is the absolute minimum of `f(x)=xlnx`?

Answer 1

Minimum Point at `(1/e, -1/e)`

Explanation:

the given `f(x) = x* ln x`

obtain the first derivative `f' (x)` then equate to zero.

`f' (x) = x*(1/x) + ln x * 1 = 0`

`1 + ln x = 0`

`ln x= -1`

`e^-1=x`

`x=1/e`

Solving for `f(x)` at `x= 1/e`

`f(x)=(1/e)*ln (1/e)`

`f(x)=(1/e)*(-1)`

`f(x)=-1/e`

so the point `(1/e, -1/e)` is located at the 4th quadrant which is a minimum point.