CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question
Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY2 Marks
Question
$\text{If }\text{ f(x)}=|\text{x}|^3,$ show that f″(x) exists for all real x and find it.
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Answer
It is known that, $|\text{x}|=\ \begin{cases}\text{x}& \quad \text{if x } \geq 0\\ -\text{x},& \quad \text{if x} <0 \end{cases}$
Therefore when $\text{x}\geq,\ \text{f(x)}=|\text{x}|^3=\text{x}^3$
In this case, $\text{f}'\text{(x)}=3\text{x}^2\text{and hence},\ \text{f}''\text{(x)}=6\text{x}$
When $\text{x}<0,\ \text{f(x)}=|\text{x}|^3=(-\text{x})^3=-\text{x}^3$
In this case, $\text{f}'\text{(x)}=-3\text{x}^2\text{ and hence},\ \text{f}''\text{(x)}=-6\text{x}$
Thus, for $\text{f(x)}=|\text{x}|^3,\ \text{f}''\text{(x)}$ exists for all real xand is given by,
$\text{f}''\text{x}=\ \begin{cases}6\text{x},& \quad \text{if x } \geq 0\\ -6\text{x},& \quad \text{if x} <0 \end{cases}$
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