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Find the smallest positive integer p for which the expression x² - 2px + 3p + 4 is negative for at least one real value of x.

1. (A) 4
2. (B) 5
3. (C) 6
4. (D) 7

Correct answer: (B) 5

Solution

The quadratic f(x) = x² - 2px + 3p + 4 has minimum value at x = p: f(p) = p² - 2p² + 3p + 4 = -p² + 3p + 4. For f to be negative somewhere, we need -p² + 3p + 4 < 0, i.e., p² - 3p - 4 > 0, i.e., (p-4)(p+1) > 0. Since p is a positive integer, this holds when p > 4, i.e., p >= 5. The smallest such p is 5.

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