Local Extrema

Loose Definition

• If $f^{\prime \prime }(c)<0$, we have a local maxima
• If $f^{\prime \prime }(c)>0$, we have a local minima

Formal Definition

For minima: $\exists \delta >0,0<|x-c|<\delta ⟹f(c)\le f(x)$ For maxima: $\exists \delta >0,0<|x-c|<\delta ⟹f(c)\ge f(x)$

Box Ball Definition

Let $f:U\subset {\mathbb{R}}^n\to \mathbb{R}$ be a function. Point $x_0$ is a Local Minima if: $\exists$ Neighborhood $x_0\in V\subset U$ s.t $x\in U⟹f(x_0)<f(x)$ Point $x_0$ is a Local Maxima if: $\exists$ Neighborhood $x_0\in V\subset U$ s.t $x\in U⟹f(x_0)>f(x)$

Not to be confused with Global Extrema

Local Maxima

Maximas are points where the surrounding slopes point downwards.

Local Minima

Minimas are points below their surrounding slopes.

Degree from # of turning points

The degree of the graph, will always have at max, (degree-1) number of turning points.