How do you find the critical points for xlnx and the local max and min?

Jun 27, 2017

The derivative of $x \ln x$ is given by the product rule.

$y ' = 1 \left(\ln x\right) + x \left(\frac{1}{x}\right)$

$y ' = \ln x + 1$

The critical points occur when the derivative equals $0$ or is undefined (the latter will only be a critical point if the point is defined in the original function).

$0 = \ln x + 1$

$- 1 = \ln x$

${e}^{-} 1 = x$

The derivative is undefined at $x = 0$, but the function is as well, so we can't count it as a critical point.

Whenever $x > {e}^{-} 1$, the derivative is positive, therefore the function is increasing. This signifies that $x = {e}^{-} 1$ is an absolute minimum.

Here is a graphical confirmation.

graph{xlnx [-18.02, 18.01, -9.01, 9.01]}

Hopefully this helps!