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Question

Which of the following function is a monotonically increasing function?


A

f(x)=x2+2x

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B

f(x)=x3+2x

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C

f(x)=x3+2x

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D

f(x)=cosx

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Solution

The correct option is B

f(x)=x3+2x


We saw that for a monotonically increasing function f’(x) 0. We will go through the options to find the answer.

a. f(x) = x2+2x

The domain of this function is R.

f’(x) = 2x+2=2(x+1)

2x+2 will be non negative or greater than equal to zero for x1. For x<1, f’(x) will be negative.

So the function will be increasing only in the interval of [1,) and will be decreasing in the rest of the domain. Thus the function is not monotonically increasing function

b. f(x) = x3+2x

The domain for this function is also R.

f’(x) = 3x2+2

3x2+20 for all real numbers.

So the function will be monotonically increasing in its domain.
Derivative of sinx would give cosx and derivative of cosx would give -sinx. Since -sinx and cosx takes both positive and negative values, we can’t say both the options C and D behaves the same way throughout their domain. So, they can’t be monotonically increasing as well.

{Note - If specifically not mentioned we’ll consider all those points in the domain where the functions exists. }


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