To find the critical points of the function defined by the derivative , set the derivative equal to zero:

This simplifies to . The first factor gives us as a solution, while the second factor has no real solutions since leads to complex numbers. Thus, the only critical point in the real number domain is at .

Next, we can examine this critical point’s nature (minimum or maximum) by evaluating the second derivative. Start by finding the second derivative:

Since for all real numbers , the function is concave up at . Hence, is a local minimum for the function .