Question

Let the area of the region \( \{(x, y) : 2y \leq x^2 + 3, \, y + |x| \leq 3, \, y \geq |x - 1|\} \) be \( A \). Then \( 6A \) is equal to:

• A. 16
• B. 18
• C. 14
• D. 12

Hint:

To find the area of a region defined by inequalities, sketch the region and use integration or geometric formulas as needed.

Answer 1

The Correct Option is D

Identify the inequalities:

• \( 2y \leq x^2 + 3 \)
• \( y + |x| \leq 3 \)
• \( y \geq |x - 1| \)

Rearrange the inequalities to express \( y \) explicitly:

• \( y \leq \frac{x^2}{2} + \frac{3}{2} \)
• \( y \leq 3 - |x| \)
• \( y \geq |x - 1| \)

Multiply the area \( A \) by 6 to find \( 6A \).

Final Answer: 6A = 12