 Differentiability
Question

# Which of the following function is thrice differentiable at x = 0?

Moderate
Solution

## $\begin{array}{l} \mathrm{f}\left(\mathrm{x}\right)=\left|{\mathrm{x}}^{3}\right|=\left\{\begin{array}{l}-{\mathrm{x}}^{3},\mathrm{x}<0\\ {\mathrm{x}}^{3}, \mathrm{x}\ge 0\end{array}\right\ \mathrm{or} \mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=\left\{\begin{array}{l}-6, \mathrm{x}<0\\ 6, \mathrm{x}>0\end{array}\right\\\ \mathrm{Hence}, \mathrm{f}\text{'}\text{'}\left(0\right) \mathrm{does} \mathrm{not} \mathrm{exist}.\\ \mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^{3}\left|\mathrm{x}\right|=\left\{\begin{array}{l}-{\mathrm{x}}^{4}, \mathrm{x}<0\\ {\mathrm{x}}^{4}, \mathrm{x}\ge 0\end{array}\right\ \mathrm{or} \mathrm{f}\text{'}\text{'}\text{'}\left(\mathrm{x}\right)=\left\{\begin{array}{l}-24\mathrm{x}, \mathrm{x}<0\\ 24\mathrm{x}, \mathrm{x}>0\end{array}\right\.\\ \mathrm{Hence}, \mathrm{f}\text{'}\text{'}\text{'}\left(0\right)=0 \mathrm{and} \mathrm{it} \mathrm{exists}.\\ \mathrm{Similarly}, \mathrm{for} \mathrm{f}\left(\mathrm{x}\right)=\left|\mathrm{x}\right|{\mathrm{sin}}^{3}\mathrm{x} \mathrm{and} \mathrm{f}\left(\mathrm{x}\right)=\mathrm{x}\left|{\mathrm{tan}}^{3}\mathrm{x}\right| \mathrm{also}, \mathrm{f}\text{'}\text{'}\text{'}\left(0\right)=0\\ \mathrm{and} \mathrm{it} \mathrm{exists}\end{array}$

Get Instant Solutions  