Question

The maximum value of the function \[ y = e^{5 + \sqrt{3} \sin x + \cos x} \]

• A. \( e^7 \)
• B. \( e^2 \)
• C. \( e^5 \)
• D. \( e^8 \)

Hint:

For functions involving trigonometric terms, maximize the argument of the exponential function to find the maximum value.

Answer 1

The Correct Option is A

Step 1: Analyzing the function.
We are given the function \( y = e^{5 + \sqrt{3} \sin x + \cos x} \). The maximum value of this function occurs when the exponent is maximized.
Step 2: Maximizing the exponent.
To maximize the exponent \( 5 + \sqrt{3} \sin x + \cos x \), we need to find the maximum value of \( \sqrt{3} \sin x + \cos x \). This is a standard trigonometric expression, and its maximum value occurs when \( \sin x = 1 \) and \( \cos x = 0 \). Substituting these values gives the maximum of \( \sqrt{3} \cdot 1 + 0 = \sqrt{3} \). Thus, the maximum value of the exponent is \( 5 + \sqrt{3} \).

Step 3: Conclusion.
The maximum value of \( y \) occurs when the exponent is maximized, which gives the result \( y = e^{5 + \sqrt{3}} \). The maximum value is \( e^7 \), corresponding to option (A).