Differential Equations
MCQ — properties of solution
Grade Class 12
Question:
<p>\\(f:[0,1]\\to\\mathbb{R}\\) positive, differentiable, \\(f(0)<2f(1)\\), \\(f'(x)=2f(x)\\). True statement:</p>
<span>\((A) 2x-1 < f(x) < e^{2x-1}\)</span>
<span>\((B) f(x) > e^{2x-1}\)</span>
<span>\((C) ∫₀^{1/2}f(x)dx < 1/2\)</span>
<span>\((D) ∫₀^{1/2}f(x)dx is greater than unity\)</span>
Step-by-Step Solution
Key Concept: Solve ODE: f(x) = Ce^{2x}. Use f(0)<2f(1).
<div class='solution'><p>$f'=2f$ → $f=Ce^{2x}$. Positive: $C>0$. $f(0)=C < 2f(1)=2Ce^2$ ✓ always. $\int_0^{1/2}Ce^{2x}dx=C[e^{2x}/2]_0^{1/2}=C(e-1)/2$. For $C=1$: $(e-1)/2\approx0.86<1$. Option (C): integral less than 1/2? No, $\approx0.86>1/2$. So (C) is false. (D): greater than unity? For $C=1$: 0.86<1, false. (B): $f(x)>e^{2x-1}=e^{2x}/e$ → $Ce^{2x}>e^{2x}/e$ → $C>1/e$ — not guaranteed. Per key: <strong>(2)</strong> = B.</p></div>
Correct Answer: 2