Question

For the function $\mathrm{f}\left(\mathrm{x}\right)=0$ for $x\le 0$, $\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^2\mathrm{logx}$ for $\mathrm{x}>0$

• A. $f’\left(0\right)$ exist at $x=0$
• B. $f’\left(0\right)$ does not exist
• C. $Lf’\left(0\right)=–1$
• D. $Rf’\left(0\right)=1$

Answer 1

The correct option is A

$f’\left(0\right)$ exist at $x=0$

Explanation for the correct option:

Finding limit on the given function.

Left hand limit,

$\begin{array}{rcl}Lf’\left(0\right)& =& f’(0-)=\frac{li{m}_{h\to 0}[f(0–h)–f(0\left)\right]}{h}\\ & =& li{m}_{h\to 0}\frac{[0–0]}{h}=0\end{array}$

Right hand limit,

$\begin{array}{rcl}Rf’\left(0\right)& =& f’(0+)=li{m}_{h\to 0}\frac{[f(0+h)–f(0\left)\right]}{h}\\ & =& li{m}_{h\to 0}\frac{[h^2\log h]}{h}\\ & =& limh\to 0h\log h=0\\ f’\left(0\right)& =& 0\end{array}$

Since left hand and right hand limit exist at $x=0$

Therefore, $f’\left(0\right)$ exist at $x=0$

Hence, correct option is option (A).