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NDA Math Application of Derivatives

For the next two items that follow :

Consider the function

f(x)=\left|x-1\right|+x^{2}

where

x\in\mathbb{R}

[NDA I 2016]

Question : 51 of 73

Marks: +1, -0

$f(x)$ is increasing in $\left(-\infty,\frac{1}{2}\right)$ and decreasing in $\left(\frac{1}{2},\infty\right)$
$f(x)$ is decreasing in $\left(-\infty,\frac{1}{2}\right)$ and increasing in $\left(\frac{1}{2},\infty\right)$
$f(x)$ is increasing in $(-\infty, 1)$ and decreasing in $(1,\infty)$
$f(x)$ is decreasing in $(-\infty,1)$ and increasing in $(1,\infty)$

Solution:

As we know

f(x)=|x-1|+x^{2}\;\;\;\;\forall\;\;\;x\in\mathbb{R}

f(x)=\begin{cases} x-1+x^2, & x\ge 1 \\ 1-x+x^2, & x<1 \end{cases}

\Rightarrow f'(x)=\begin{cases} 2x+1\text{;}, & x\ge 1 \\ 2x-1\text{;}, & x<1 \end{cases}

f(x)

is in quadratic form (parabola).Hence

f(x)

is decreasing in

\left(-\infty,\frac{1}{2}\right)

and increasing

\left(\frac{1}{2},\infty\right)

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