GATE | GATE CS 2020 | Question 11

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 deceasing 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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