JEE Advanced 2021 Paper 2 Q15 Mathematics Differentiation & Applications Mean Value Theorems Hard

JEE Advanced 2021 Paper 2 · Q15 · Mean Value Theorems

Which of the following statements is TRUE?

  1. A. $f(\sqrt{\ln 3}) + g(\sqrt{\ln 3}) = \dfrac{1}{3}$
  2. B. For every $x > 1$, there exists an $\alpha\in(1,x)$ such that $\psi_1(x) = 1 + \alpha x$
  3. C. For every $x > 0$, there exists a $\beta\in(0,x)$ such that $\psi_2(x) = 2x(\psi_1(\beta) - 1)$
  4. D. $f$ is an increasing function on the interval $\left[0, \dfrac{3}{2}\right]$
Reveal answer + step-by-step solution

Correct answer:C

Solution

$\psi_2'(x) = 2x - 2 + 2e^{-x} = 2(x + e^{-x} - 1) = 2(\psi_1(x) - 1)$. By LMVT on $\psi_2$ on $[0,x]$: $\dfrac{\psi_2(x) - \psi_2(0)}{x - 0} = \psi_2'(\beta)$ for some $\beta\in(0,x)$. Since $\psi_2(0) = 0$, we get $\psi_2(x) = x\cdot\psi_2'(\beta) = x\cdot 2(\psi_1(\beta)-1) = 2x(\psi_1(\beta)-1)$. Option (C).

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