Question

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

Answer 1

The derivative of `xlnx` is given by the product rule.

`y' = 1(lnx) + x(1/x)`

`y' = lnx + 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 = lnx + 1`

`-1 = lnx`

`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!