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 ) (13,43) 1 3 , 4 3 (12,32) 1 2 , 3 2 (23,53) 2 3 , 5 3 (1,2) ( 1 , 2 )
To determine where the function is decreasing throughout the interval, we need to find where the derivative of the function is negative. Taking the derivative of the function:
f'(x) = -πsin(πx) - π√sin(πx)cos(πx)
To find the intervals where the function is decreasing, we need to find where the derivative is negative. Since the derivative involves trigonometric functions, it may be difficult to find exact values.
However, we can analyze the sign of the derivative in different intervals. By looking at the signs of sin(πx) and cos(πx) in different intervals, we can determine where the derivative is negative.
For example, in the interval (0,1), sin(πx) and cos(πx) are positive, meaning -πsin(πx) and -π√sin(πx)cos(πx) are also negative. Therefore, f'(x) is negative in this interval.
By analyzing the signs of sin(πx) and cos(πx) in the other given intervals, we find that f'(x) is negative in the interval (0,1) and (1,2).
Thus, the correct answer is: (0,1) and (1,2).