Find the intervals on which the given function is increasing and decreasing.
y=-x^2-2x
y'=-2x-2
critical point
x=-1
Increasing
[-infinity,-1)
Decreasing
(-1,infinity]
is this the correct way of writting this wiht the brackets
you got it right.
good job.
so the parenthasis means including and the bracket means its open or how does this work
Yes, the way you have written the intervals using brackets is correct.
To determine the intervals on which the given function is increasing and decreasing, you need to analyze the sign of the derivative.
In this case, the derivative of the function is y' = -2x - 2. To find the critical points, you set the derivative equal to zero and solve for x.
-2x - 2 = 0
-2x = 2
x = -1
The critical point is x = -1.
Now, to determine if the function is increasing or decreasing on the intervals, you can use the first derivative test.
When the derivative is positive (greater than zero), the function is increasing.
When the derivative is negative (less than zero), the function is decreasing.
For this function, the derivative is y' = -2x - 2.
Substitute test values into the derivative:
For x < -1, choose x = -2:
y' = -2(-2) - 2 = 4 - 2 = 2 (positive)
This means the function is increasing on the interval (-infinity, -1).
For x > -1, choose x = 0:
y' = -2(0) - 2 = -2 (negative)
This means the function is decreasing on the interval (-1, infinity).
Therefore, your intervals are correctly written as:
Increasing interval: (-infinity, -1)
Decreasing interval: (-1, infinity]
The use of brackets [ and ] indicates that the endpoint is included in the interval, while the use of parentheses ( and ) indicates that the endpoint is not included in the interval.