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### Course: AP®︎/College Calculus AB>Unit 5

Lesson 2: Extreme value theorem, global versus local extrema, and critical points

# Critical points introduction

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.
• 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?
• Absolutely! Wonderful observation btw.
• What is the difference between global and local maximum?
• A global maximum is the highest point in the entire function's range, while a local maximum is the highest point in a specific area.
• 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.
• Why does Sal say that the critical numbers of a function exclude endpoints of a closed interval?
• Because derivatives aren't very well-defined at the endpoint of a closed interval.
• 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.
• Is a critical point the same as a stationary point?
• The definition of a critical point is one where the derivative is either 0 or undefined. A stationary point is where the derivative is 0 and only zero. Therefore, all stationary points are critical points (because they have a derivative of 0), but not all critical points are stationary points (as they could have an undefined derivative).
• I was wondering, for the graph y = 3, the slope is always 0. Does that mean that graph has an infinite amount of critical points?
• Interesting question. So it seems.

I was looking for the definition of critical value and I found this:

''Any value of `x` for which `f'(x) = 0` or undefined is called a critical value for `f`''. (Source:http://bit.ly/1Aq2x8e, page 106).

If we apply that definition to a derivative of a constant, then it has an infinite amount of critical points/values.

And because the function fails to pass the First Derivative Test, there are no extrema.
(1 vote)