How do you find the intervals of increasing and decreasing given #y=(3x^23)/x^3#?
The intervals of decreasing are
The intervals of increasing are
To determine the intervals of increasing and decreasing, we compute the derivative.
We have a fraction of two functions here.
So,
So,
We can now create our sign chart.
plot{(3x^23)/(x^3) [10, 10, 5, 5]}
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To find the intervals of increasing and decreasing for the function ( y = \frac{{3x^2  3}}{{x^3}} ), follow these steps:
 Find the derivative of the function.
 Determine where the derivative is positive (increasing) and where it is negative (decreasing).
 Use test points to confirm the sign of the derivative in each interval.
 State the intervals of increasing and decreasing accordingly.
Let's proceed with these steps:

The derivative of the function ( y = \frac{{3x^2  3}}{{x^3}} ) can be found using the quotient rule, which states that if ( u(x) ) and ( v(x) ) are differentiable functions, then the derivative of ( \frac{{u(x)}}{{v(x)}} ) is ( \frac{{u'(x)v(x)  u(x)v'(x)}}{{[v(x)]^2}} ).

After finding the derivative, determine where it is positive or negative to identify the intervals of increasing and decreasing.

Use test points within each interval to confirm the sign of the derivative.

State the intervals of increasing and decreasing based on the sign of the derivative.
If you'd like, I can provide the steps for finding the derivative and proceed with the calculations.
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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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