# How do you find the critical points of a rational function?

*To find the critical points of a function, first ensure that the function is differentiable, and then take the derivative. Next, find all values of the function's independent variable for which the derivative is equal to 0, along with those for which the derivative does not exist. These are our critical points.
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*The critical points of a function #f(x)# are those where the following conditions apply:*

A) The function exists.

B) The derivative of the function

As an example with a polynomial function, suppose I take the function

For our first type of critical point, those where the derivative is equal to zero, I simply set the derivative equal to 0. Doing this, I find that the only point where the derivative is 0 is at

For our second type of critical point, I look to see if there are any values of

For a slightly more tricky example, we will take the function

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To find the critical points of a rational function, follow these steps:

- Identify the rational function, typically in the form ( f(x) = \frac{p(x)}{q(x)} ), where ( p(x) ) and ( q(x) ) are polynomials.
- Find the derivative of the function ( f'(x) ).
- Set ( f'(x) ) equal to zero and solve for ( x ). These solutions are potential critical points.
- Check the critical points by evaluating the second derivative ( f''(x) ) at each critical point.
- Determine whether each critical point corresponds to a local maximum, local minimum, or neither based on the sign of ( f''(x) ).
- Exclude any critical points where the second derivative test is inconclusive.
- The remaining critical points are the critical points of the rational function.

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*Answer from HIX Tutor*

*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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