Question

If \( B = \sin^2y + \cos^4y \), then for all real \( y \),

• A. \( \frac{3}{4} \leq B \leq 1 \)
• B. \( \frac{3}{4} \leq B \leq \frac{13}{16} \)
• C. \( \frac{13}{16} \leq B \leq 1 \)
• D. \( 1 \leq B \leq 2 \)

Hint:

To find the range of a trigonometric expression, analyze the behavior of individual terms within their known bounds.

Answer 1

The Correct Option is A

Step 1: Given the expression for \( B \): \[ B = \sin^2y + \cos^4y \] Step 2: To determine the range of \( B \), we first know that \( \sin^2y \) lies between 0 and 1, and \( \cos^4y \) also lies between 0 and 1. Therefore, the sum of these two terms will also lie between 0 and 1. Step 3: The minimum value of \( \sin^2y \) is 0, and the maximum value of \( \cos^4y \) is 1. Thus, the minimum value of \( B \) is \( \frac{3}{4} \) and the maximum value is 1. Thus, the range of \( B \) is: \[ \frac{3}{4} \leq B \leq 1 \]