For which of the following intervals is the function f(x)=cos(πx)+3–√sin(πx) decreasing throughout the interval? (1 point) Responses (0,1) ( 0 , 1 ) (1/3,4/3) 1 3 , 4 3 (1/2,3/2) 1 2 , 3 2 (2/3,5/3) 2 3 , 5 3 (1,2) ( 1 , 2 )
To find when the function is decreasing, we need to find the intervals where the derivative of the function is negative. Let's find the derivative of f(x).
f(x) = cos(πx) + 3 – √sin(πx)
f'(x) = -πsin(πx) - (π/2)(1/√sin(πx))*(πcos(πx))
= -πsin(πx) - (π/2)(πcos(πx))/√sin(πx))
= -πsin(πx) - (π^2/2)(cos(πx))/√sin(πx)
To find the critical points, we set the derivative equal to zero and solve for x:
-πsin(πx) - (π^2/2)(cos(πx))/√sin(πx)) = 0
Since sin(πx) cannot be zero (sin(πx) = 0 when x is an integer), we can multiply both sides of the equation by √sin(πx) to eliminate the denominator:
-πsin^2(πx) - (π^2/2)(cos(πx)) = 0
Now let's analyze the sign of the derivative on different intervals. We need to consider the intervals between the critical points and the endpoints of the given intervals.
On the interval (0,1):
-πsin(0) - (π^2/2)(cos(0))/√sin(0)) = 0 - (π^2/2)(1)/√0) = 0 - 0 = 0
Since the derivative is zero on this interval, f(x) is not decreasing throughout the interval (0,1).
On the interval (1/3,4/3):
-πsin(1/3) - (π^2/2)(cos(1/3))/√sin(1/3)) < 0
-πsin(4/3) - (π^2/2)(cos(4/3))/√sin(4/3)) < 0
Since the derivative is negative on this interval, f(x) is decreasing throughout the interval (1/3,4/3).
On the interval (1/2,3/2):
-πsin(1/2) - (π^2/2)(cos(1/2))/√sin(1/2)) < 0
-πsin(3/2) - (π^2/2)(cos(3/2))/√sin(3/2)) < 0
Since the derivative is negative on this interval, f(x) is decreasing throughout the interval (1/2,3/2).
On the interval (2/3,5/3):
-πsin(2/3) - (π^2/2)(cos(2/3))/√sin(2/3)) < 0
-πsin(5/3) - (π^2/2)(cos(5/3))/√sin(5/3)) < 0
Since the derivative is negative on this interval, f(x) is decreasing throughout the interval (2/3,5/3).
On the interval (1,2):
-πsin(1) - (π^2/2)(cos(1))/√sin(1)) = 0 - (π^2/2)(-1)/√1) = π^2/2
-πsin(2) - (π^2/2)(cos(2))/√sin(2)) = 0 - (π^2/2)(1)/√0) = 0 - 0 = 0
Since the derivative is positive on this interval, f(x) is not decreasing throughout the interval (1,2).
Therefore, the function is decreasing throughout the intervals (1/3, 4/3), (1/2, 3/2), and (2/3, 5/3).
The correct answer is (1/3, 4/3), (1/2, 3/2), and (2/3, 5/3).