How do you find the intervals of increasing and decreasing given #y=(3x^2-3)/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^2-3)/(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:
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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}} ).
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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 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.
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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