Multiple choice

The equation $4y^2 - 3y + C = 0$ has real roots. The value of C for which the product of the roots is a maximum is

  1. $\frac{9}{16}$
  2. $\frac{9}{4}$
  3. $\frac{4}{9}$
  4. $\frac{3}{4}$
  5. $-\frac{4}{3}$
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A Correct answer
Explanation

For the equation 4y^2 - 3y + C = 0 to have real roots, the discriminant D = (-3)^2 - 4(4)(C) = 9 - 16C must be >= 0, so C <= 9/16. The product of the roots is C/4. Since this is an increasing function of C, the maximum value occurs at the largest possible value of C, which is 9/16.

AI explanation

By Vieta's formulas, the product of the roots for the equation 4y^2 - 3y + C = 0 is C divided by 4. For the roots to be real, the discriminant must be non-negative, meaning (-3)^2 - 4(4)(C) >= 0, or 9 - 16C >= 0. This limits C to a maximum value of 9 divided by 16. The maximum product of the roots occurs at this boundary, which is 9 divided by 64, but evaluating the options against the condition reveals that the maximum allowable C yielding real roots is 9 divided by 16.

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