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GATE | GATE CS 2020 | Question 11

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Consider the functions I. e^{-x} II. x^{2} - \sin x III. \sqrt{x^3+1} Which of the above functions is/are increasing everywhere in [0, 1] ? (A) Ⅲ only (B) Ⅱ only (C) Ⅱ and Ⅲ only (D) Ⅰ and Ⅲ only

Answer: (A)

Explanation: If the derivative of a function is positive in given domain, then it is increasing function, else decreasing function. So, I.  \frac{\partial }{\partial \:x}\left(e^{-x}\right) = -e^{-x} II.  \frac{\partial }{\partial \:x}\left(x^2-\sin\:x\right) = 2x-\cos \left(x\right) III.  \frac{\partial }{\partial \:x}\left(\sqrt{x^3+1}\right) = \frac{3x^2}{2\sqrt{x^3+1}}  Therefore, only (III) is increasing everywhere in [0, 1]. Option (A) is correct.

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Last Updated : 17 Dec, 2021
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