Local Extrema

Not to be confused with Global Minima And Maxima

Local Maxima

Maximas are points where the surrounding slopes point downwards.

Loose Definition

If $f^{\prime \prime }(c)<0$

Formal Definition

$\exists \delta >0,0<|x-c|<\delta ⟹f(c)\ge f(x)$

Local Minima

Minimas are points below their surrounding slopes.

Loose Definition

If $f^{\prime \prime }(c)>0$

Formal Definition

$\exists \delta >0,0<|x-c|<\delta ⟹f(c)\le f(x)$ Minimas are points below their surrounding slopes.

Finding leading degree from # of turning points

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

in our graph above, there are 3 turningo points. This gives us lots of options, but we know at least that it is at least a degree 4 or higher graph.

Pair this with a second, or third method of determining powers like Endpoint Behavior or Finite Differences to find out the actual leading degree of the graph