GATE Computer Science (CS) 2020 Solved Papers

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Question : 32 of 65

Marks: +1, -0

Consider the functions
I.

{e}^{- {x}}

II.

x^{2}-\sin x

III.

√{x^{3}+1}

Which of the above functions is/are increasing everywhere in

[0,1] ?

II and III only
I and III only
II only
III only

Solution:

(i)

{e}^{-x}

is decreasing
(ii)

f(x)=x^{2}-\sin x

f^{'}(x)=2 x-\cos x

Now check

f^{'}(x)

at

[0,1]

f^{'}(0)=-1

f^{'}(1)=2-\cos 1=1.4596

∴ f(x)

is not increasing everywhere
(iii)

f(x)=√{x^{3}+1}

f^{'}(1)=\;{1}/{2 √{x^{3}+1}} × 3 x^{2}

{f}^{'}( {x}) ≥ 0

for all

{x}

.

∴ {f}( {x})

is an increasing function.

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