# What do critical points tell you?

A function's critical point, also known as a "local" extreme or extreme value, is the location where the function may have a relative extremum.

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Critical points in mathematics typically refer to points where the derivative of a function is either zero or undefined. These points provide important information about the behavior of the function. Specifically:

- Critical points can indicate where a function has maximum, minimum, or saddle points.
- They help in identifying potential local extremums (maximum or minimum values) of the function.
- They can be used to analyze the concavity of the function and determine inflection points.
- Critical points are crucial for optimization problems as they represent potential solutions.
- They aid in sketching the graph of the function by highlighting key features such as peaks, valleys, and points of inflection.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- What are the extrema of # f(x)=1/x^3 +10x# on the interval [1,6]?
- How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #y = x^3 - 3x^2 - 9x +15#?
- What are the extrema of #g(x) = 2 sin(2x - pi) + 4# on #[-pi/2,pi/2]#?
- What are the critical values of #f(x)=3x^2+2x+5#?
- What are the critical values, if any, of # f(x)= sin^2x +cos^2x in [0,oo]#?

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