Question

Let \( f(x) = 2 - 7 \sin{\left( \frac{2x}{7} \right)} \). Then the maximum value of \( f(x) \) is:

• A. -5
• B. 5
• C. 4
• D. 9
• E. -9

Hint:

To find the maximum or minimum values of trigonometric functions, remember that \( \sin{x} \) has a maximum value of \( 1 \) and a minimum value of \( -1 \). Use these limits to compute the maximum and minimum of the function.

Answer 1

The Correct Option is D

The function \( f(x) \) is given by: \[ f(x) = 2 - 7 \sin{\left( \frac{2x}{7} \right)}. \] The maximum value of \( \sin{\theta} \) is \( 1 \), so the maximum value of \( -7 \sin{\left( \frac{2x}{7} \right)} \) is \( -7 \times (-1) = 7 \). 
Thus, the maximum value of \( f(x) \) occurs when \( \sin{\left( \frac{2x}{7} \right)} = -1 \), and is: \[ f(x) = 2 + 7 = 9. \] Thus, the maximum value of \( f(x) \) is \( 9 \), which corresponds to option (D).