Unconstrained Maximization

1. The first-order necessary condition for a maximum or minimum. If a function has a maximum or minimum at a certain point, then its derivative at that point is zero

   • If $f(x)$ has a maximum or minimum at $x=c$, then $f^{\prime }(c)=0$.

2. The second-order necessary condition for a maximum. If a function has a maximum at a certain point, then its second derivative at that point is less than or equal to zero:

   • If $f(x)$ has a maximum at $x=c$, then $f^{\prime \prime }(c)\le 0$.

3. The second-order sufficient condition for a maximum. If a function’s first derivative is zero at a certain point and its second derivative at that point is less than zero, then the function has a maximum at that point:

   • If $f^{\prime }(c)=0$ and $f^{\prime \prime }(c)<0$, then $f(x)$ has a maximum at x=c.

4. To find the unconstrained maximum of a function, we can set its derivative equal to zero and solve for ‘x’. Then, we use the second derivative test to determine whether each solution is a maximum or minimum.

   • Find unconstrained maximum of $f(x)$
     • set $f^{\prime }(x)=0$ and solve for ‘x’.
     • Use $f^{\prime \prime }(x)$ to verify maxima.