Find the interval(s) where the function is increasing where and it is decreasing.
f(x)=sin(x+(pi/2)) for 0≤x≤2pi
so my derivative is f'(x)=cos(x+(pi/2))?
and my critical number is -pi/2?
cosine is positive for x +pi/2 = 0 to pi/2 and from 3 pi/2 to 2 pi
when x+pi/2 = 0 then x = -pi/2 or 3pi/2
when x + pi/2 = pi/2, x = 0
so one interval is x = 3pi/2 to 0
when x+pi/2 = 3 pi/2 then x = pi
when x+pi/2 = 2 pi then x = 3 pi/2
so second interval is frm x = pi to 3 pi/2
so finally the entire interval from x = pi to x = 2 pi (which is 0) has a positive derivative, function increasing
Wow, so confusing. Thanks!
So the whole interval is increasing? The function of cosx has a section of decreasing though.
To find the interval(s) where the function is increasing or decreasing, we first need to find the derivative of the function and identify the critical numbers.
The given function is f(x) = sin(x + (π/2)) for 0 ≤ x ≤ 2π.
To find the derivative, we differentiate f(x) with respect to x. The derivative of sin(x) is cosine, so the derivative of f(x) is f'(x) = cos(x + (π/2)).
Now, we need to find the critical numbers by setting the derivative f'(x) equal to zero and solving for x:
cos(x + (π/2)) = 0
We know that cos(0) = 1, and the cosine function has a period of 2π, so we can solve the equation within the interval [0, 2π].
Adding -(π/2) to both sides:
x + (π/2) = -(π/2)
x = -(π/2) - (π/2)
x = -π
So, the critical number is x = -π.
Now, let's analyze the intervals.
1. Interval (0, -π):
To determine if the function is increasing or decreasing in this interval, we can evaluate the derivative f'(x). Substitute a value within this interval, such as x = -π/2:
f'(-π/2) = cos(-π/2 + (π/2))
= cos(0)
= 1
Since f'(-π/2) = 1, which is positive, the function is increasing in the interval (0, -π).
2. Interval (-π, 2π):
Similarly, let's evaluate the derivative f'(x) within this interval. Substitute a value within, such as x = π/2:
f'(π/2) = cos(π/2 + (π/2))
= cos(π)
= -1
Since f'(π/2) = -1, which is negative, the function is decreasing in the interval (-π, 2π).
Hence, the interval where the function f(x) = sin(x + (π/2)) is increasing is (0, -π), and the interval where it is decreasing is (-π, 2π).