Match List-I with List-II

Choose the correct answer from the options given below:

(A)-(IV), (B)-(III), (C)-(II), (D)-(I)

The correct answer is Option (2) → (A)-(IV), (B)-(III), (C)-(II), (D)-(I)

$\text{(A)}\ f(x)=x|x|=\begin{cases}x^{2},& x\ge0\\-x^{2},& x<0\end{cases}$

$f'(x)=\begin{cases}2x,& x>0\\-2x,& x<0\end{cases}\Rightarrow f'(x)>0\ \forall x\ne0$

$\Rightarrow$ increases on $(-\infty,\infty)$ $\Rightarrow$ (IV).

$\text{(B)}\ f(x)=x^{2}+2x-5=(x+1)^{2}-6$

$f'(x)=2x+2$; $f'(x)<0$ for $x<-1$ $\Rightarrow$ decreases on $(-\infty,-1)$ $\Rightarrow$ (III).

$\text{(C)}\ f(x)=x^{2}-6x+9=(x-3)^{2}$

$f'(x)=2x-6$; $f'(x)>0$ for $x>3$ $\Rightarrow$ increases on $(3,\infty)$ $\Rightarrow$ (II).

$\text{(D)}\ f(x)=-x^{2}$

$f'(x)=-2x<0$ for $x>0$ $\Rightarrow$ decreases on $(0,\infty)$ $\Rightarrow$ (I).

Matching: (A)→(IV), (B)→(III), (C)→(II), (D)→(I)