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

f(x+y)=f(x)⋅f(y)⇒f(x)=eλxf′(0)=4a⇒f′(x)=λeλx⇒λ=4aSo, fx=e4axf′′(x)−3af′(x)−f(x)=0⇒λ2−3aλ−1=0⇒16a2−12a2−1=0⇒4a2=1⇒a=12F(x)=e2x Area =∫02 exdx=e2−1

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

f(x+y)=f(x)⋅f(y)⇒f(x)=eλxf′(0)=4a⇒f′(x)=λeλx⇒λ=4aSo, fx=e4axf′′(x)−3af′(x)−f(x)=0⇒λ2−3aλ−1=0⇒16a2−12a2−1=0⇒4a2=1⇒a=12F(x)=e2x Area =∫02 exdx=e2−1

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Let ƒ : R → R be a twice differentiable function such that ƒ(x + y) = ƒ(x) ƒ(y) for all x, y $\in$ R. If ƒ'(0) = 4a and ƒ staisfies ƒ''(x) – 3a ƒ'(x) – ƒ(x) = 0, a > 0, then the area of the region $R = {(x, y) ∣ 0 \leq y \leq f (ax), 0 \leq x \leq 2}$ is :

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e⁴+ 1

e⁴- 1

e²+ 1

e²- 1

answer is A.

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$\begin{aligned} f (x + y) = f (x) \cdot f (y) \end{aligned}$

$\begin{array}{r} \Rightarrow f (x) = e^{λ x} f^{'} (0) = 4 a \end{array}$

$\Rightarrow f^{'} (x) = λ e^{λ x} \Rightarrow λ = 4 a$

So, $f (x) = e^{4 a x}$

$\begin{aligned} f^{''} (x) - 3 {af}^{'} (x) - f (x) = 0 \end{aligned}$

$\begin{array}{r} \Rightarrow λ^{2} - 3 a λ - 1 = 0 \end{array}$

$\Rightarrow 16 a^{2} - 12 a^{2} - 1 = 0 \Rightarrow 4 a^{2} = 1 \Rightarrow a = \frac{1}{2}$

$\begin{aligned} F (x) = e^{2 x} \end{aligned}$

$Area = \int_{0}^{2} e^{x} d x = e^{2} - 1$

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

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Let ƒ : R → R be a twice differentiable function such that ƒ(x + y) = ƒ(x) ƒ(y) for all x, y $\in$ R. If ƒ'(0) = 4a and ƒ staisfies ƒ''(x) – 3a ƒ'(x) – ƒ(x) = 0, a > 0, then the area of the region $R = {(x, y) ∣ 0 \leq y \leq f (ax), 0 \leq x \leq 2}$ is :