# How do you find the critical points of the function #f(x) = x / (x^2 + 4)#?

Find those points at which the derivative of

The critical points of a function are those points where its first derivative is 0, i.e. those points where the function reaches a maximum, a minimum, or a point of inflection.

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To find the critical points of the function ( f(x) = \frac{x}{x^2 + 4} ), follow these steps:

- Find the derivative of the function ( f(x) ) with respect to ( x ), denoted as ( f'(x) ).
- Set ( f'(x) ) equal to zero and solve for ( x ).
- The solutions obtained in step 2 are the critical points of the function.

Let's go through these steps:

- The derivative ( f'(x) ) can be found using the quotient rule:

[ f'(x) = \frac{(x^2 + 4) \cdot 1 - x \cdot (2x)}{(x^2 + 4)^2} ]

Simplifying this expression yields:

[ f'(x) = \frac{x^2 + 4 - 2x^2}{(x^2 + 4)^2} ] [ f'(x) = \frac{-x^2 + 4}{(x^2 + 4)^2} ]

- Set ( f'(x) ) equal to zero and solve for ( x ):

[ \frac{-x^2 + 4}{(x^2 + 4)^2} = 0 ]

Since the numerator can't be zero (it's always ( -x^2 + 4 )), the critical points occur when the denominator is zero:

[ x^2 + 4 = 0 ]

This equation has no real solutions, so there are no critical points for the function ( f(x) = \frac{x}{x^2 + 4} ).

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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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