Final answer:
To find the exact values of x such that f(x) = 0 for the given function, we solve the trigonometric equation by setting the function equal to zero. We find possible solutions where the cosine equals 1/2, and then check which of these are within the interval [4, 14].
Step-by-step explanation:
To find all exact values for x in the interval [4, 14] such that f(x) = 0 for the function f(x) = 144 cos (\(\frac{\pi}{6}\) (x + 1)) - 72, we need to solve the equation for x. Setting the function equal to zero gives us:
144 cos (\(\frac{\pi}{6}\) (x + 1)) - 72 = 0
Add 72 to both sides:
144 cos (\(\frac{\pi}{6}\) (x + 1)) = 72
Divide both sides by 144:
cos (\(\frac{\pi}{6}\) (x + 1)) = \(\frac{72}{144}\) = \(\frac{1}{2}\)
Using our knowledge of cosine, we know that cos(\(\frac{\pi}{3}\)) = \(\frac{1}{2}\). Thus, we look for angles where the cosine is \(\frac{1}{2}\), in the form of:
\(\frac{\pi}{6}\) (x + 1) = \(\frac{\pi}{3}\) + k2\pi
Or:
\(\frac{\pi}{6}\) (x + 1) = -\(\frac{\pi}{3}\) + k2\pi
Where k is an integer. By solving these equations, we find x = 4, 7, 10, and 14 as possible solutions within the given interval. However, we must verify each solution to ensure it's within the provided constraints and makes the function equal zero.