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Consider the functions

  1.  $e^{-x}$
  2. $x^{2}-\sin x$
  3. $\sqrt{x^{3}+1}$

Which of the above functions is/are increasing everywhere in  $[ 0,1]$?

  1. Ⅲ only
  2. Ⅱ only
  3. Ⅱ and  Ⅲ only
  4. Ⅰ and Ⅲ only
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3 Answers

Best answer
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74 votes

Answer A. III Only   (i.e. $\sqrt{x^3 + 1}$ )

Explanation

Decreasing/Increasing nature of a function can be determined by observing the first derivative of equations in given domain.

If the derivative is positive in given domain, It is increasing, else a negative value indicates it is decreasing.

Now testing one by one

I. $e^{-x}$

Here, $\frac{\mathrm{d} }{\mathrm{d} x}e^{-x} = -e^{-x}$

This will remain negative in entire domain [0,1] hence decreasing.

II. $x^{2} - \sin(x)$

Here, $\frac{\mathrm{d} }{\mathrm{d} x}(x^{2} - \sin(x)) = 2x - \cos(x)$

Here the switch happens! Observe $2x - \cos(x)$ is negative for $x=0$ (i.e. = -1) and positive at $x=1$ (i.e. = ~1.4596977). Though we can find the exact point where it switches but that's not required here. We can confirm that till some point in the domain [0,1], this function decreases and then increases.

III. $\sqrt{x^3 + 1}$

Here, (Though this one is intuitive), $\frac{\mathrm{d} }{\mathrm{d} x}\sqrt{x^3 + 1} = \frac{3x^{2}}{2\sqrt{x^3+1}}$   This will remain positive throughout [0,1] and hence will be increasing.

edited by
5 votes
5 votes
A. 3 only .

1. e^-x is popular decreasing function.

2. if you will put 0.5 ( used virtual calculator) you will get some -ve value in x^2 - sinx . at 0 function value is 0 at 0.5 it is -ve it means not increasing function.

3. very easy to tell increasing function.
1 votes
1 votes
III Only.

Increasing fun means:For all values of x within the domain of the fun,

Increasing in x will result increasing in f(x)

&&

Dcreasing in x will result decreasing in f(x).

Now,for option I,

x=0----->f(x)=1;x=1----->f(x)=0.368

[Increasing x does not result increasing f(x)]

For option II,

X=0----->f(x)=0;x=0.5----->f(x)=0.25-sin(180*0.5/3.14)=-0.22

[Increasing x does not result increasing f(x)]

Note that value of x is in radian,so u have converted in degree to use it inside sin function.

For option III,

Clearly f(x) increases/decreases whenever u increase/decrease x within [0,1].
Answer:

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