Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f$ be a differentiable function on $R$ and satisfying $f(x) = -(x^2 - x + 1)e^x + \int_0^x e^{xy} f'(y)dy$. If $f(1) + f'(1) + f''(1) = ke$, where $k \in N$, then find $k$.
Step-by-Step Solution
Key Concept: Differentiate the functional equation to eliminate the integral and solve for $f(x)$ explicitly.
Given the functional equation $f(x) = e^x \int_0^x e^{-y} f'(y) dy - (x^2 - x + 1)e^x$, differentiate both sides to obtain $f'(x) = f'(x) + \int_0^x e^{x-y} f'(y) dy - e^x(x^2 + x)$. This simplifies to $0 = f(x) + (x^2 + x + 1)e^x - e^x(x^2 + x)$, giving $f(x) = e^x(x^2 - x - 1)$. Computing derivatives: $f'(x) = e^x(2x - 1)$, $f''(x) = 2e^x + (2x-1)e^x + 2e^x$. Therefore $f(1) + f'(1) + f''(1) = e + 2e + 5e = 9e$ and $k = -9$.
Correct Answer: 9