Answer:
Concave up on the intervals (-π/2, π/2), and concave down on the intervals (-π,-π/2) and (π/2,π).
Explanation:
To find the intervals where the function y = -cot(x) is concave up and concave down, we need to find the second derivative of the function and then determine its sign for different intervals.
First, we find the first derivative of y = -cot(x):
dy/dx = csc^2(x)
Then, we find the second derivative by differentiating the first derivative:
d^2y/dx^2 = -2csc^2(x) * cot(x)
We know that the function is concave up when the second derivative is positive, and concave down when the second derivative is negative.
Now, we need to determine the intervals where the second derivative is positive and negative.
d^2y/dx^2 > 0 when:
-2csc^2(x) * cot(x) > 0
csc^2(x) < 0 and cot(x) < 0 or csc^2(x) > 0 and cot(x) > 0
Since csc^2(x) is always positive, we can ignore the first inequality. Therefore, the second derivative is positive when cot(x) > 0.
d^2y/dx^2 < 0 when:
-2csc^2(x) * cot(x) < 0
csc^2(x) > 0 and cot(x) < 0 or csc^2(x) < 0 and cot(x) > 0
Since csc^2(x) is always positive, we can ignore the first inequality. Therefore, the second derivative is negative when cot(x) < 0.
Recall that cot(x) = cos(x)/sin(x), which changes signs at x = kπ, where k is an integer.
So, we can now use this information to determine the intervals of concavity:
For x in (-π/2,0) and (0,π/2), cot(x) is positive and therefore, the function is concave up.
For x in (-π,-π/2) and (π/2,π), cot(x) is negative and therefore, the function is concave down.
Therefore, the function y = -cot(x) is concave up on the intervals (-π/2, π/2), and concave down on the intervals (-π,-π/2) and (π/2,π).