Problem: Given the equation 3x2−4x+k=0 with real roots. The value of k for which the product of the roots of the equation is a maximum, is:
Answer Choices:
A. 916
B. 316
C. 94
D. 34
E. −34
Solution:
The product of two numbers whose sum is a fixed quantity is maximized when each of the numbers is one-half the sum. Since the sum of the roots is 34, the maximum product, 94, is obtained when each of the roots is 32. Therefore, k/3=94, so that k=34.
or
Solving the given equation for k/3 (the product of the roots), we have k/3=34x−x2=94−(32−x)2. The right side of this equation is a maximum when x=32, so that k/3 is a maximum when x=32. ∴k/3(max)=34⋅32−94=94.∴k(max)=34
or
The product of the roots is k/3. We seek the largest possible k consistent with real roots. For real roots 16−12k≧0, so that 34≧k. Hence, the desired k=34.