Im having a hard time figuring out this problem:
Use the function, y=-(x+1)^2 +2, to answer the following parts.
The increasing interval and decreasing interval
To find both I believe that:
increasing : x < -1
decreasing : x > 1
So is this on solution below
Increasing interval (-4,-7)
Decreasing interval(2,-7)
The increasing and decreasing intervals depend on the second derivative:
f'(x)=dy/dx;=-2(x+1)
If f'(x)>0, the function is increasing.
If f'(x)<0, the function is decreasing.
Since f'(x)=0 at x=-1,
f'(x)>0 for (-∞-1], and
f'(x)<0 for [-1,∞)
The increasing and decreasing intervals can be deduced.
Note that x=-1 is included in both intervals. If this is not clear why, look up the definition of increasing and decreasing intervals, or post.
Thank you for your help.
You're welcome!
Sorry, there is a typo in my response, although it does not affect the results.
The first sentence should read:
"The increasing and decreasing intervals depend on the first derivative:... "
To determine the increasing and decreasing intervals of the function y = -(x+1)^2 + 2, you need to analyze the behavior of the function with respect to the x-values.
The function y = -(x+1)^2 + 2 represents a downward opening parabola, since the coefficient in front of the squared term is negative. This means that the function is decreasing as x increases and increasing as x decreases.
To find the increasing interval, you need to identify the range of x-values where the function is increasing. This occurs when the graph of the function is sloping upwards. In other words, you need to find the x-values where the slope is positive.
For the given function, the slope is positive when x < -1. Therefore, the increasing interval is x < -1.
To find the decreasing interval, you need to identify the range of x-values where the function is decreasing. This occurs when the graph of the function is sloping downwards. In other words, you need to find the x-values where the slope is negative.
For the given function, the slope is negative when x > -1. Therefore, the decreasing interval is x > -1.
Therefore, the correct solution for the increasing interval is x < -1, and the correct solution for the decreasing interval is x > -1.
The solution you provided:
Increasing interval (-4,-7)
Decreasing interval (2,-7)
is not correct. To determine the correct intervals, it is essential to understand the behavior of the function and analyze the slope.