Final answer:
To find the critical numbers of the function g(θ) = 28θ - 7 tan(θ) on the interval (0, 2π), we need to find the values of θ where the derivative equals zero or is undefined.
Step-by-step explanation:
To find the critical numbers of the function on the interval (0, 2π), we need to find the values of θ where g'(θ) = 0 or does not exist. The critical numbers occur when the derivative of g(θ) equals zero or is undefined. Let's find the derivative of g(θ) first:
g'(θ) = 28 - 7(sec(θ))^2
Now, we can set g'(θ) equal to zero and solve for θ:
28 - 7(sec(θ))^2 = 0
7(sec(θ))^2 = 28
(sec(θ))^2 = 4
sec(θ) = ±2
Since sec(θ) = 1/cos(θ), we can rewrite the equation as:
1/cos(θ) = ±2
cos(θ) = ±1/2
Now, we need to find the values of θ for which the cosine is equal to ±1/2. These occur at θ = π/3, 5π/3, and 2π/3, 4π/3.
Therefore, the critical numbers of the function g(θ) on the interval (0, 2π) are π/3, 5π/3, 2π/3, and 4π/3.