Final answer:
To find the value of f(Ï/12), we first determine the amplitude (A) and the vertical shift (D) by using the maximum and minimum values of the function provided. After finding A = 10 and D = 12, we substitute these values and x = Ï/12 into the function and calculate f(Ï/12) to be 17.
Step-by-step explanation:
The function given is f(x)= Asin(2x) + D, where both A and D are positive numbers. The maximum value of f(x) is 22, and the minimum value is 2. To find the values of A and D, we use the fact that the maximum value of sine function (sin(2x)) is 1 and the minimum is -1. Therefore, the amplitude (A) plus D must equal the maximum value, while the negative amplitude minus D must equal the minimum value.
Max value: A + D = 22
Min value: -A + D = 2
We can solve these two equations simultaneously to find A and D.
A + D = 22
-A + D = 2
Adding both equations, we get:
2D = 24
D = 12
Substituting D in one of the equations:
A + 12 = 22
A = 10
Now we can find f(Ï/12) by substituting A, D, and x into the original function:
f(Ï/12) = 10sin(2 * Ï/12) + 12
f(Ï/12) = 10sin(Ï/6) + 12
f(Ï/12) = 10 * 0.5 + 12
f(Ï/12) = 5 + 12
f(Ï/12) = 17
Therefore, the value of f(Ï/12) is B) 17.