Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
In the interval $(a,b)$ there exists at least one point $c$, for any two differentiable function $f$ and $g$ such that $\begin{vmatrix} f(a) & f(b) \\ \phi(a) & \phi(b) \end{vmatrix} - \lambda^2(b-a)\begin{vmatrix} f(a) & f'(c) \\ \phi(a) & \phi'(c) \end{vmatrix}$, then sum of absolute value of $\lambda$ is_____.
Step-by-Step Solution
Key Concept: Functional equations are solved by using boundary conditions and differentiation to extract the derivative, then integrating.
From the functional equation $f(x+y) = f(x) + f(y) + xy(x+y)$, setting $y=0$ gives $f(0) = 0$. Computing $\lim_{h \to 0} \frac{f(h)}{h} = -1$ and differentiating the functional equation yields $f'(x) = -1 + x^2$. Integrating: $f(x) = -x + \frac{x^3}{3} + c$. Since $f(0) = 0$, we have $c = 0$, so $f(x) = \frac{x^3}{3} - x$. Finally, $f(3) = 9 - 3 = 6$ (note: the solution shows $3^2 - 1 = 8$, indicating verification of differentiability).
Correct Answer: 2