Final answer:
To find the maximum and minimum values of the function g(theta) = 4theta - 7 sin (theta), we need to take the derivative, set it equal to zero, and solve for theta. The maximum value of the function occurs at theta = theta_0 + 2npi, while the minimum value occurs at theta = theta_0 + (2n+1)pi. Without knowing the value of theta_0, we cannot determine the specific maximum and minimum values.
Step-by-step explanation:
To find the maximum and minimum values of the function g(theta) = 4theta - 7 sin (theta), we need to find the critical points by taking the derivative of the function and setting it equal to zero. Let's start by taking the derivative:
g'(theta) = 4 - 7 cos (theta)
Setting g'(theta) equal to zero, we can solve for theta:
4 - 7 cos (theta) = 0
7 cos (theta) = 4
cos (theta) = 4/7
Using the unit circle or a calculator, we can find the angles at which cosine is equal to 4/7. Let's call this angle theta_0.
The maximum value of the function occurs at theta = theta_0 + 2npi, where n is an integer. The minimum value of the function occurs at theta = theta_0 + (2n+1)pi, where n is an integer.
Therefore, the maximum and minimum values of the function g(theta) = 4theta - 7 sin (theta) depend on the value of theta_0. Without knowing the exact value of theta_0, we cannot determine the specific maximum and minimum values of the function. So, none of the given options (a), (b), (c), or (d) are correct.