JEE Mains · Maths · STD 12 - 8. Application and integration
Let \(f: \mathbf{R} \rightarrow \mathbf{R}\) be a twice differentiable function such that \(f(x+y)=f(x) f(y)\) for all \(x, y \in \mathbf{R}\). If \(f^{\prime}(0)=4 \mathrm{a}\) and \(f\) satisfies \(f^{\prime \prime}(x)-3 \mathrm{a} f^{\prime}(x)-f(x)=0\), \(\mathrm{a}\gt0\), then the area of the region \(\mathrm{R}=\{(x, y) \mid 0 \leq y \leq f(\mathrm{a} x), 0 \leq x \leq 2\}\) is:
- A \(e^2-1\)
- B \(e^2+1\)
- C \(e^4+1\)
- D \(e^4-1\)
Answer & Solution
Correct Answer
(A) \(e^2-1\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & f(x+y)=f(x) \cdot f(y) \\ & \Rightarrow f(x)=e^{\lambda x} f^{\prime}(0)=4 a \\ & \Rightarrow f^{\prime}(x)=\lambda e^{\lambda x} \Rightarrow \lambda=4 a \end{aligned}\) So, \(f(x)=e^{4 x}\)…
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