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## AP®︎/College Calculus BC

### Unit 5: Lesson 2

Extreme value theorem, global versus local extrema, and critical points

# Critical points introduction

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.C (LO)
,
FUN‑1.C.1 (EK)
,
FUN‑1.C.2 (EK)
,
FUN‑1.C.3 (EK)
Sal introduces the "critical points" of a function and discusses their relationship with the extremum points of the function. Created by Sal Khan.

## Want to join the conversation?

• at : Why is f'(x) not defined?
• I imagine the tangent line to that point as being a vertical line as there is no other way to draw it that does not intersect with another part of the graph. If you wanted to find the slope of that tangent line it would be undefined because a vertical line has an undefined slope. This is because the x values are the same making the change in x ( x2-x1) equal to zero. Therefore because division by zero is undefined the slope of the vertical tangent line is undefined.
• Wouldn't x0 (x naught) also be a local maximum?
• You are correct. x_0 is both the global max and a local max.
• Can the global min/max also be called the absolute min/max?
• Yes. And Local min/max can also be called relative min/max.
• So being a critical point is the necessary, but not sufficient, condition for being minima / maxima? And being a minima / maxima is the sufficient, but not necessary, condition for being a critical point?
• What is the difference between global and local maximum?
• at why did he located local minimum there, the smallest value of y in this graph where curve cuts the x-axis, why not this point can't be local minimum.
• First you have to understand the definition of local minimum and global minimum.

The global minimum is the lowest value for the whole function.

The local minimum is just locally. Visually this means that it is decreasing on the left and increasing on the right. The y values just a bit to the left and right are both bigger than the value. This also means the slope will be zero at this point. It is a transitioning phase. If all these things are true then its a local minimum.

As for your point that is not a local minimum because it keeps on decreasing.
• Can the global maximum be a local maxima as well?
• The global maximum must be a local maximum. It is just afforded a special name because there is one global maximum while there can be several local maxima.
• Is local minimum the same as the relative minimum and global maximum the same as absolute maximum??
• Yes. local = relative and absolute = global.