Question

Find the critical numbers of the function on the interval (0, 2π). g(θ) = 28θ − 7 tan θ

Answer 1

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.