Let f: {\R} \rightarrow R be a twice differentiable function such that f(x+y)=f(x) f(y) for all x, y \in R. If f^{'}(0)=4 a and f satisfies f^{'}(x)-3 a f(x)-f(x)=0, a>0, then the area of the region R=\{(x, y) ∣ 0 \leq y \leq f(a x), 0 \leq x \leq 2\} is: [22 Jan 2025 Shift 1]

Differential Equations Part 3

Section: Mathematics

Question : 18 of 34

Marks: +1, -0

Let f:R⟶R be a twice differentiable function such that f(x+y)=f(x)f(y) for all x,y∈R. If f′(0)=4a and f satisfies f′(x)−3af(x)−f(x)=0,a>0, then the area of the region R={(x,y)∣0≤y≤f(ax), 0≤x≤2} is:

[22 Jan 2025 Shift 1]

e2−1
e4−1
e2+1
e4+1

Validate

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