Critical Value/Number

The x value c of point (in the domain of f) in a function f that in the domain of f such that either f’(c)=0 or f’(c)=DNE.

Critical point

If x= c is a critical value, then (c, f(c)) is a critical point.

<aside> 🔑 Remember!!! the theorem only suggest those points as a POTENTIAL position of an extrema. Not guarantee!

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Theorem for Absolute Extrema

Definition of Absolute (Global) Extrema

Absolute/ Global Extrema can only occur at a critical value or at an endpoint of an interval.

• Critical value of f’(c)=0, there may a change of direction of the graph making a peak/valley of the graph (no more extreme in the local position)
• Critical value of f’(c)=DNE
  • there is a “ jump increase/decrease” of the graph, as the locals didn’t keep up to. (hole but a point that have jump from the normal graph)
  • there is cusp, corner, etc…
• The endpoint is possible to have the points to be less extreme than itself: see the graph if confused

Theorem for Local Extrema

Absolute/ Global Extrema can only occur at a critical value of an open

(?-?|,there can be local extrema in a close interval, just view them as open interval?) interval

• Critical value of f’(c)=0, there may a change of direction of the graph making a peak/valley of the graph (no more extreme in the local position)
• Critical value of f’(c)=DNE
  • there is a “ jump increase/decrease” of the graph, as the locals didn’t keep up to. (hole but a point that have jump from the normal graph)
  • there is cusps, corner, etc… (not end points for no points in the other side.)