Question

The difference between greatest and least value of \[ f(x) = 2 \sin x + \sin 2x, \, x \in \left[ 0, \frac{3\pi}{2} \right] \] is

• A. \( \frac{3}{5} \)
• B. \( \frac{3}{2} \)
• C. \( \sqrt{5} \)
• D. \( \frac{3}{2} \)

Hint:

To find the range of a trigonometric function, take its derivative and use critical points to determine the maximum and minimum values.

Answer 1

The Correct Option is C

Step 1: Find the maximum and minimum values of the function.
To find the maximum and minimum values, use the derivative of \( f(x) = 2 \sin x + \sin 2x \) to determine the critical points.
Step 2: Conclusion.
Thus, the difference between the maximum and minimum values of \( f(x) \) is \( \sqrt{5} \).
Final Answer: \[ \boxed{\sqrt{5}} \]