JEE Advanced 2021 Paper 2 Q16 Mathematics Integration & Differential Equations Definite Integrals Hard

JEE Advanced 2021 Paper 2 · Q16 · Definite Integrals

Which of the following statements is TRUE?

  1. A. $\psi_1(x)\le 1$, for all $x > 0$
  2. B. $\psi_2(x)\le 0$, for all $x > 0$
  3. C. $f(x)\ge 1 - e^{-x^2} - \dfrac{2}{3}x^3 + \dfrac{2}{5}x^5$, for all $x\in\left(0, \dfrac{1}{2}\right)$
  4. D. $g(x)\le \dfrac{2}{3}x^3 - \dfrac{2}{5}x^5 + \dfrac{1}{7}x^7$, for all $x\in\left(0, \dfrac{1}{2}\right)$
Reveal answer + step-by-step solution

Correct answer:D

Solution

Using the series $e^{-t} = 1 - t + \dfrac{t^2}{2} - \dfrac{t^3}{6} + \cdots$, we get $\sqrt{t}\,e^{-t} \le t^{1/2} - t^{3/2} + \dfrac{t^{5/2}}{2}$ for small $t$ (alternating series upper bound). Actually a cleaner upper bound: $\sqrt{t}\,e^{-t} \le \sqrt{t}(1 - t + \tfrac{t^2}{2})$. Integrating on $[0, x^2]$: $g(x) = \int_0^{x^2}\sqrt{t}\,e^{-t}\,dt \le \dfrac{2}{3}x^3 - \dfrac{2}{5}x^5 + \dfrac{1}{7}x^7$ for $x\in(0,1/2)$. Option (D).

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