<p>Find the number of points on the curve \(y^2 = x + \sin x\) where the tangent is horizontal, given that \(|y| \leq 3\).</p>
Step-by-Step Solution
Key Concept: For horizontal tangents, set dy/dx = 0. The constraint |y| ≤ 3 limits the valid x values.
<p><strong>Step 1:</strong> Differentiate the curve equation: $y^2 = x + \sin x$</p><p>$2y \frac{dy}{dx} = 1 + \cos x$</p><p><strong>Step 2:</strong> For horizontal tangent, $\frac{dy}{dx} = 0$</p><p>This requires $1 + \cos x = 0$, so $\cos x = -1$</p><p>Therefore $x = (2n+1)\pi$</p><p><strong>Step 3:</strong> Apply constraint $0 \leq x \leq 9$ and $|y| \leq 3$:</p><p>$0 \leq (2n+1)\pi \leq 9$ gives $n = 0$, so $x = \pi$</p><p><strong>Step 4:</strong> At $x = \pi$: $y^2 = \pi + \sin\pi = \pi$, so $y = \pm\sqrt{\pi}$</p><p>The points are $(\pi, \sqrt{\pi})$ and $(\pi, -\sqrt{\pi})$</p><p>∴ Number of points is <strong>2</strong>.</p>
Correct Answer: 2