Solution assignment 09 Optimizing f(x)

Verify whether the function:

$y=x\ln(x)$

has a maximum or a minimum.

We calculate the derivative (product rule) and make it equal to $0$ :

$y'=1\cdot\ln(x)+x\cdot\displaystyle\frac{1}{x}=\ln(x)+1=0$

This equation has the solution:

$x=\displaystyle\frac{1}{e}=e^{-1}$

In order to verify whether we have a maximum or a minimum, we calculate the second derivative:

$y''=\displaystyle\frac{1}{x}=e>0$

For $x=e^{-1}$ there is a minimum with the coordinates: $(e^{-1},-e^{-1})$