Integral Calculus-1
Integral Calculus-1
Allen Star Batch
Grade 12
Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be a function satisfying $f(x+2y) = f(x)e^{2y} + f(2y)e^x + x^2\left(1-e^{2y}\right) + 4y^2\left(1-e^x\right) + 4xyy$ for all $x, y \in \mathbb{R}$ and $f'(0) = 1$, then:
$f(x) = xe^x + x^2$
$f(x) = xe^x - x^2$
$\int f(x)dx = e^x(x-1) - \frac{x^3}{3} + c$
$\int f(x)dx = e^x(x-1) - \frac{x^3}{3} + c$
Step-by-Step Solution
Key Concept: Use the functional equation with strategic substitutions (y=0, then differentiation with respect to y) to extract f'(x), then integrate using the boundary condition f(0)=0 and f'(0)=1 to determine f(x)=xe^x+x^2.
Given the functional equation $f(x+2y) = f(x)e^{2y} + f(2y)e^x + x^2(1-e^{2y}) + 4y^2(1-e^x) + 4xy$, set $x = y = 0$ to obtain $f(0) = 2f(0)$, so $f(0) = 0$. To find $f'(x)$, compute $\lim_{y \to 0} \frac{f(x+2y) - f(x)}{2y}$ by expanding $e^{2y}$ and $e^x$ as series and simplifying the resulting expression.
Correct Answer: 1,3