GATE 2025 ST – Question 11
Differentiable f:[0,∞)→[0,∞) has f(0)=0,f(x)>0 for x>0, and $f(x)^2=\int_0^x[f(t)^2+f(t)]dt$. Which range contains f(2)?
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Correct answer: (B) 1<f(2)≤2
Explanation
Differentiation gives 2ff′=f²+f. Since f>0 for x>0, f′=(f+1)/2. Solve with continuity at zero: f(x)=e^(x/2)−1. Thus f(2)=e−1≈1.718, giving **B**.