# How do you find the intervals of increasing and decreasing using the first derivative given #y=(x-1)^2(x+3)#?

The function

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chart{(x-1)^2(x+3) [-5, 5, -10, 10]}

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To find the intervals of increasing and decreasing for the function (y = (x - 1)^2(x + 3)), we first find the first derivative, (y'), then determine where it is positive (increasing) or negative (decreasing).

Given (y = (x - 1)^2(x + 3)), let's first find (y'):

[y = (x - 1)^2(x + 3)] [y' = 2(x - 1)(x + 3) + (x - 1)^2]

Now, to find where (y') is positive or negative, we find the critical points by setting (y') equal to zero and solving for (x):

[0 = 2(x - 1)(x + 3) + (x - 1)^2]

Solve for (x) to find critical points. Then, use the first derivative test to determine the intervals of increasing and decreasing.

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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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