How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for #f(x)=(x^3)/(x^24)#?
The intervals of increasing are
The intervals of decreasing are
We calculate the first derivative and construct a sign chart.
We need
We construct the sign chart
graph{x^3/(x^24) [14.24, 14.24, 7.12, 7.12]}
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To determine where the function ( f(x) = \frac{x^3}{x^2  4} ) is increasing or decreasing and to find relative maxima and minima, follow these steps:
 Find the derivative of the function, ( f'(x) ).
 Set ( f'(x) = 0 ) and solve for critical points.
 Use the first derivative test to determine where the function is increasing or decreasing around the critical points.
 Find the second derivative, ( f''(x) ), and use it to determine the concavity of the function.
 Use the second derivative test to identify relative maxima and minima.
Now, let's go through these steps:

Find the derivative: [ f(x) = \frac{x^3}{x^2  4} ] [ f'(x) = \frac{d}{dx}\left(\frac{x^3}{x^2  4}\right) ]

Set ( f'(x) = 0 ) and solve for critical points.

Use the first derivative test to determine where the function is increasing or decreasing around the critical points.

Find the second derivative: [ f'(x) = \frac{d^2}{dx^2}\left(\frac{x^3}{x^2  4}\right) ]

Use the second derivative test to identify relative maxima and minima.
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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