Single-variable optimization, results

Summary

First derivative test (necessary conditions for extreme values)

• $f:A→B$ and $x_0∈A$ . If $f$ is differentiable at $x_0$ and $x_0$ is a local interior extreme point then $x_0$ must be a stationary point.
• Examples:
• If $f(x)=x^2$ (natural domain $R$ ) then $f$ is differentiable everywhere and all points are interior points. The extreme point $x_0=0$ must be a stationary point, $f'(0)=0$ .
• If $f(x)=x$ on $[0,1]$ then $x_0=0$ is an extreme point. Since $x_0$ is not an interior point, it need not be a stationary point (and it is not).
• If $f(x)=|x|$ then $x_0=0$ is an interior extreme point but since $f$ is not differentiable at $x=0$ it need not be a critical point (and it is not).
• The opposite is not true. If $f(x)=x^3$ then $f$ is differentiable but the interior stationary point $x=0$ is not a local extreme point (it is a saddle point).

Second derivative test

• $f:A→B$ and $x_0∈A$ . $f$ is twice differentiable at $x_0$ and $x_0$ is a stationary point.
• If $f''(x_0)<0$ then $x_0$ is a strict local maximum point
• If $f''(x_0)>0$ then $x_0$ is a strict local minimum point
• If $f''(x_0)=0$ then no conclusion may be drawn.
• Examples:
• If $f(x)=x^2$ then $x_0=0$ is a stationary point. Since $f^{''}\left( 0 \right)=2>0$ , it is a strict local minimum point.
• If $f(x)=-x^2$ then $x_0=0$ is a stationary point. Since $f^{''}\left( 0 \right)=-2<0$ , it is a strict local maximum point.
• If $f(x)=x^3$ then $x_0=0$ is a stationary point. Since $f^{''}\left( 0 \right)=0$ , the second derivative test is inconclusive. Since $f$ is strictly increasing, $x_0=0$ is not a local extreme point. An interior stationary point which is not a local extreme point is called a saddle point .
• If $f(x)=x^4$ then $x_0=0$ is a stationary point. Since $f^{''}\left( 0 \right)=0$ , the second derivative test is inconclusive. Since $f\left( 0 \right)=0$ and $f\left( x \right)>0$ if $x≠0$ , $x_0=0$ is a a strict local minimum point.
• If $f\left( x \right)=-x^4$ then $x_0=0$ is a stationary point. Since $f^{''}\left( 0 \right)=0$ , the second derivative test is inconclusive. Since $f\left( 0 \right)=0$ and $f\left( x \right)<0$ if $x≠0$ , $x_0=0$ is a a strict local maximum point.
• If $f(x)=1$ then all points are stationary points. Since $f''(0)=0$ for all $x$ the second derivative test is inconclusive. In this case, every point is a local minimum and a local maximum point, although not strict.

Extreme value theorem

• If $f$ is continuous function and the domain of $f$ is a closed and bounded interval, $I=[a,b]$ then there exists a global maximum point $x_{max}$ in $I$ and a global minimum point $x_{min}$ in $I$ .
• If $y_{max}=f\left( x_{max} \right)$ and $y_{min}=f\left( x_{min} \right)$ then $y_{min}≤f(x)≤y_{max}$ for all $x∈I$ .
• $x_{max}$ and $x_{min}$ need not be unique. If $f(x)=2$ on $I=\left[ 0,1 \right]$ then every point in $[0,1]$ is a maximum point and a minimum point.
• $f$ need not be differentiable for the extreme value theorem to hold.

Concave and convex functions

• If $f$ is convex function defined on an interval $I$ and $x_0$ is a stationary point, then $x_0$ is a global minimum point.
• If $f$ is concave function defined on an interval $I$ and $x_0$ is a stationary point, then $x_0$ is a global maximum point.
• If $f$ is strictly convex / concave then $x_0$ is a strict global minimum / maximum point.