Differential Calculus
Differential Calculus
star_batch_jee_advanced_2025
Grade 12
Question:
Let $f_1(x)$ and $f_2(x)$ be twice differentiable function. Where $F(x) = f_1(x) + f_2(x)$ and $G(x) = f_1(x) - f_2(x)$, $\forall x \in \mathbb{R}$. $f_1(0) = 2$ and $f_2(0) = 1$. If $f_1'(x) = f_2(x)$ and $f_2'(x) = f_1(x)$, $\forall x \in \mathbb{R}$, then the number of solutions of the equation $(F(x))^2 = \frac{9x^4}{G(x)}$ is______.
Step-by-Step Solution
Key Concept: Substituting the given functions into the equation reduces it to a transcendental equation whose number of solutions is found by counting intersections.
Given $F(x) = 3e^x$ and $G(x) = e^{-x}$, the equation $9x^4 = [F(x)]^2 G(x)$ becomes $9x^4 = (3e^x)^2 \cdot e^{-x} = 9e^{2x-x} = 9e^x$, which simplifies to $x^4 = e^x$. This transcendental equation can be analyzed graphically or numerically to determine intersections between $y = x^4$ and $y = e^x$.
Correct Answer: 2