<p><strong>765.</strong> If \(f(x) = (x - a)(x - b)\) for \(a, b \in R\), then find the minimum number of roots of equation \[\pi(f'(x))^2 \cos(\pi(f(x))) + \sin(\pi(f(x))) f''(x) = 0\] in \([\alpha, \beta]\) where \(f(\alpha) = 3 = f(\beta)\) and \(\alpha < a < b < \beta\).</p>
Step-by-Step Solution
Key Concept: Recognize that the given equation can be rewritten as the derivative of a composite function: d/dx[sin(πf(x))] = 0, which means sin(πf(x)) has critical points. Combined with f(α) = f(β) = 3, we need to count zeros of this derivative in [α,β].
<p><strong>Step 1:</strong> Rewrite the given equation. Notice that π(f'(x))² cos(πf(x)) + sin(πf(x))·f''(x) = d/dx[sin(πf(x))] by the chain rule, since d/dx[sin(πf(x))] = πf'(x)cos(πf(x))·f'(x) + sin(πf(x))·f''(x) = π(f'(x))²cos(πf(x)) + f''(x)sin(πf(x)).</p><p><strong>Step 2:</strong> Define g(x) = sin(πf(x)). We need g'(x) = 0 in [α,β].</p><p><strong>Step 3:</strong> Evaluate g at endpoints: g(α) = sin(π·3) = sin(3π) = 0 and g(β) = sin(π·3) = sin(3π) = 0.</p><p><strong>Step 4:</strong> Since f(x) = (x-a)(x-b) is a parabola with vertex between a and b, and f(α) = f(β) = 3 with α < β, by Rolle's theorem applied to f(x), there exists at least one critical point c ∈ (α,β) where f'(c) = 0.</p><p><strong>Step 5:</strong> Between any two zeros of g(x) where g is continuous, there must be at least one zero of g'(x). Since g(α) = g(β) = 0 and g is continuous on [α,β], by Rolle's theorem, there is at least one point where g'(x) = 0.</p><p><strong>Step 6:</strong> Since f is a parabola with one critical point and f(α) = f(β), the function f must have a minimum in (α,β). The equation g'(x) = 0 has roots wherever f'(x) = 0 (if cos(πf) ≠ 0) or at points where f''(x) = 0 (which is nowhere since f''(x) = 2 ≠ 0). At the minimum point where f'(c) = 0, if cos(πf(c)) ≠ 0, then this is a root.</p><p><strong>Step 7:</strong> Minimum analysis: f has exactly one critical point in (α,β). The constraint that g(α) = g(β) = 0 with g differentiable guarantees at least <strong>one root</strong> by Rolle's theorem.</p><p>∴ Answer: <strong>1</strong></p>
Correct Answer: 1