How do you find the intervals of increasing and decreasing given #y=x^3+2x^2+2#?
Well, the slope of the equation's graph at each point is represented by the first derivative.
Wherever this is greater than zero, the function is increasing.
X can be factored out:
These prove to be the interval's endpoints where the function is increasing, but you can further analyze the data:
Because a negative number times a negative number is positive, 3x is > 0, so the term (3x + 4) must be > 0 if x is less than 0. Additionally, any positive number plus 4 is also positive.
Given that a negative number (keep in mind that x < 0) times a positive number is always negative, x (3x+ 4) must therefore be NEGATIVE.
Thus, in the interval where x < 0, x is DECREASING.
Examine the term (3x + 4) once more now.
Alternatively, you could graph the function and use visual aids to identify the increasing interval.
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To find the intervals of increasing and decreasing for the function y = x^3 + 2x^2 + 2, you need to analyze the sign of its derivative.

Find the derivative of the function y = x^3 + 2x^2 + 2. y' = 3x^2 + 4x

Set the derivative equal to zero and solve for x to find critical points. 3x^2 + 4x = 0 x(3x + 4) = 0 x = 0 or x = 4/3

Create a sign chart for the derivative using the critical points. Test a value in each interval created by the critical points (∞, 0), (0, 4/3), and (4/3, ∞) into the derivative.
For x < 0, choose x = 1: y' = 3(1)^2 + 4(1) = 3  4 = 1 (negative) For 0 < x < 4/3, choose x = 1: y' = 3(1)^2 + 4(1) = 3 + 4 = 1 (positive) For x > 4/3, choose x = 2: y' = 3(2)^2 + 4(2) = 12 + 8 = 4 (negative)
 Determine the intervals of increasing and decreasing. The function is increasing when the derivative is positive and decreasing when the derivative is negative.
Interval of increasing: (0, 4/3) Interval of decreasing: (∞, 0) and (4/3, ∞)
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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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