# How do you find critical points on a graph?

Alternatively, the tangent line is vertical and there are cusps and discontinuities (jump or removable) where the tangent line does not exist.

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To find critical points on a graph, follow these steps:

- Determine the derivative of the function.
- Set the derivative equal to zero and solve for (x). These points are potential critical points.
- Determine the second derivative of the function.
- Evaluate the second derivative at each potential critical point.
- If the second derivative is positive at a potential critical point, the function has a local minimum at that point.
- If the second derivative is negative at a potential critical point, the function has a local maximum at that point.
- If the second derivative is zero or undefined at a potential critical point, further analysis is needed (such as using the first derivative test or higher-order derivative tests).
- Critical points may also occur at points where the derivative is undefined (such as vertical tangents) or where the function has cusps or corners.

These steps help identify critical points, which are points where the derivative of the function is zero or undefined, and they are essential for analyzing the behavior of functions and determining features such as local extrema and inflection points.

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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.

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- What are the critical points of #f(x) = -(2x - 2.7)/ (2 x -7.29 )^2#?
- Is #f(x)=(x+2)(2x-3)(x-2)# increasing or decreasing at #x=0#?
- In which interval the function #f(x)=sqrt(x^2+8)-x# is a decreasing function?

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