Question

Find the maximum value of \( y = \min\left(\frac{1}{2} - \frac{3x^2}{4}, \frac{5x^2}{4}\right) \) for \( 0<x<1 \)

• A. 1/3
• B. 1/2
• C. 5/27
• D. 5/16

Hint:

To maximize a min expression, find intersection of curves to determine the maximum point.

Answer 1

The Correct Option is D

We are to maximize: \[ y = \min\left(\frac{1}{2} - \frac{3x^2}{4}, \frac{5x^2}{4}\right) \] Let: - \( f_1(x) = \frac{1}{2} - \frac{3x^2}{4} \) - \( f_2(x) = \frac{5x^2}{4} \) Find point where both are equal: \[ \frac{1}{2} - \frac{3x^2}{4} = \frac{5x^2}{4} \frac{1}{2} = \frac{8x^2}{4} = 2x^2 x^2 = \frac{1}{4} x = \frac{1}{2} \] Now plug back \( x = \frac{1}{2} \) into either function: \[ y = \frac{5x^2}{4} = \frac{5}{4} \times \frac{1}{4} = \frac{5}{16} \] \[ \boxed{\frac{5}{16}} \]