<p>Let <em>f</em>(<em>x</em>) be a continuous function in (0, 1) satisfying \(\int_0^1 x\sqrt{x}\, f(x)(1 - \sqrt{x}f(x))\, dx = \dfrac{1}{8}\). Number of solutions of the equation \(f(x) = e^x\) is:</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand has the form g(g') where g = √x·f(x), allowing substitution u = √x·f(x) to transform the constraint into ∫u(1-u)du = 1/8, which determines that √x·f(x) must equal a constant value (specifically 1/2) almost everywhere.
<p><strong>Step 1:</strong> Let u = √x·f(x), so the constraint becomes:</p><p>∫₀¹ u(1-u)·(du/d(√x)) dx = 1/8</p><p><strong>Step 2:</strong> Rewrite the integral: ∫₀¹ u(1-u) du = ∫₀¹ (u - u²) du = [u²/2 - u³/3]₀¹ = 1/2 - 1/3 = 1/6</p><p><strong>Step 3:</strong> This suggests testing if √x·f(x) = constant. If √x·f(x) = c (constant), then ∫₀¹ x·c(1-c) dx = c(1-c)·[x²/2]₀¹ = c(1-c)/2 = 1/8</p><p><strong>Step 4:</strong> Solving c(1-c)/2 = 1/8 gives c(1-c) = 1/4, so c² - c + 1/4 = 0, yielding c = 1/2</p><p><strong>Step 5:</strong> Therefore √x·f(x) = 1/2, which means f(x) = 1/(2√x)</p><p><strong>Step 6:</strong> For f(x) = eˣ, we need 1/(2√x) = eˣ. At x = 0⁺, LHS → ∞ while RHS → 1. At x = 1, LHS = 1/2 ≈ 0.5 while RHS = e ≈ 2.718. The LHS is decreasing and RHS is increasing, so no intersection exists.</p><p>∴ Answer: <strong>B (0 solutions)</strong></p>
Correct Answer: B