If the equation x \cos \theta = x^{2}+p has a real solution for <strong>EVERY</strong> \theta where 0 \leq \theta \leq \frac{\pi}{4}, then which one of the following is correct?

Explanation

For the quadratic equation x^2 - x \cos \theta + p = 0 to have real roots, discriminant D = \cos^2 \theta - 4p \geq 0, so 4p \leq \cos^2 \theta. This must hold for all \theta \in [0, \pi/4]. The minimum value of \cos^2 \theta in this interval is at \theta = \pi/4, which is 1/2. Thus, 4p \leq 1/2 \implies p \leq 1/8.

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