Which statement among the following is true?( | vedclass

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(A) $(i)$ Given $f(x) = x|x|$. For $x > 0$,$f(x) = x^2$,so $f'(x) = 2x > 0$. For $x  0$. Thus,$f'(x) > 0$ for all

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$(i)$

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(A) $(i)$ Given $f(x) = x|x|$. For $x > 0$,$f(x) = x^2$,so $f'(x) = 2x > 0$. For $x  0$. Thus,$f'(x) > 0$ for all $x \in R - \{0\}$,meaning $f(x)$ is strictly increasing. This statement is true.$(ii)$ Given $f(x) = \log_{1/4} (x)$. Since the base $1/4 $(iii)$ $A$ one-one function can be strictly increasing,strictly decreasing,or neither (e.g.,$f(x) = 1/x$ is one-one but not monotonic). This statement is false.$(iv)$ Given $f(x) = x^{1/3}$. $f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3x^{2/3}} > 0$ for all $x 
eq 0$. Thus,it is strictly increasing on $R$. This statement is false.

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