Question

Local Extrema Find the local maximum and minimum values of the function and the value of x at which each occurs.

State each answer rounded to two decimal places.

$g\left(x\right)=x^x,x>0$

Short answer

There is no local maximum.

The local minimum value $=0.69,\mathrm{at}x=0.37$.

Step-by-step solution

Step 1. Given.

The given function is $g\left(x\right)=x^x,x>0$.

Step 2. To determine.

We have to find the local maximum and minimum values of the function and the value of x at which each occurs.

Step 3. Calculation.

We’ll use a graphing calculator to graph $g\left(x\right)=x^x,x>0$.

From the graph, the local minimum is at $\left(0.368,0.692\right)\approx \left(0.37,0.69\right)$.

It means, the local minimum is $0.69\mathrm{at}x=0.37$.

From the graph, there is no local maximum.