Applications of Derivatives
Differential Calculus-2
star_batch_jee_advanced_2025
Grade 12
Question:
If $f: R \to R$ is a monotonic, differentiable real valued function, $a, b$ are two real numbers and $\int_{a}^{b}(f(x) + f(a))(f(x) - f(a))dx = k\int_{f(a)}^{f(b)} x(b - f^{-1}(x))dx$, then the value of $k$ is ________.
Step-by-Step Solution
Key Concept: Integration by substitution combined with inverse function properties and algebraic manipulation yields the constant relating the two integrals.
Since $f(x)$ is monotonic, $f^{-1}(x)$ exists. Let $f^{-1}(x) = z$, so $x = f(z)$. Then $z = f^{-1}(f(b)) = b$ gives $dx = f'(z)dz$. Computing $\int_{f(b)}^{f(a)} x(b-f^{-1}(x))dx = -\frac{1}{2}(b-a)(f(a))^2 + \frac{1}{2}\int_a^b (f(z))^2dz = \frac{1}{2}\int_a^b [f(z) - f(a)](f(z) + f(a))dx$. Therefore $k = 2$.
Correct Answer: 2