Final answer:
The average value of the function f(x) = 9sinx + 5cosx over the interval [0, 14π/6] is found by using the periodic properties of sine and cosine functions. The value is 7, which results from averaging the coefficients in front of the sine and cosine over one complete cycle.
Step-by-step explanation:
To find the average value of the function f(x) = 9sinx + 5cosx on the interval [0, 14π/6], we can use the fact that sin and cos are periodic functions with a period of 2π. This means that we can find the average value over one complete cycle of both sine and cosine functions, spanning 2π, and apply it to the given interval. The average value of a function f(x) over the interval [a, b] is given by:
∫ab f(x) dx / (b - a)
Since we know that the average of sin²(x) and cos²(x) over a complete cycle is 1/2, we can state that the average value of f(x) is half the sum of the coefficients in front of the sine and cosine, which yields:
Average value = (9/2 + 5/2) = 7.