Question

Given $f\left(x\right)=\left\{\begin{array}{c}{\log }_a{\left(a|\left[x\right]+[-x]|\right)}^x\frac{a^{\frac{2}{\left[\frac{\left[x\right]+[-x]}{\left|x\right|}\right]}-5}}{3+a^{\frac{1}{\left|x\right|}}}for\left|x\right|\ne 0,a>1\\ 0forx=0\end{array}\right\}$
where $[.]$ represents the integral part function, then :

• A. $f$ is continuous but not differentiable at $x=0$
• B. $f$ is continuous and differentiable at $x=0$
• C. The differentiability of $f$ at $x=0$ depends on the value of $a$
• D. $f$ is continuous and differentiable at $x=0$ and for $a=e$ only

Answer 1

Answer: (A)

$f$ is continuous but not differentiable at $x=0$

Given:$f\left(x\right)=\{\begin{array}{c}{log}_a{\left(a|\left[x\right]+[-x]|\right)}^x\frac{a^{\frac{2}{\left[\frac{\left[x\right]+[-x]}{\left|x\right|}\right]}-5}}{3+a^{\frac{1}{\left|x\right|}}}\\ 0forx=0\end{array}for\left|x\right|\ne 0,a>1$

To find: continuity and differentiability of $f\left(x\right)$ at x=0

Sol:$\left[x\right]+[-x]=\{\begin{array}{c}-1\text{x is not an integer}\\ 0\text{when x is integer}\end{array}$

$\therefore {lim}_{x\to 0^{+}}{log}_a{\left(a|-1|\right)}^x\frac{a^{\frac{2}{\left[\frac{-1}{\left|x\right|}\right]}-5}}{3+a^{\frac{1}{\left|x\right|}}}$

$=x.\left({log}_{a^a}\right).\frac{a^{\frac{2}{\left[\frac{-1}{\left|x\right|}\right]}}}{3+a^{\frac{1}{\left|x\right|}}}.a^{-5}$

$=0$

${lim}_{x\to 0^{-}}{\log }_a{\left(a|-1|\right)}^x.\frac{a^{\frac{2}{\left[\frac{-1}{\left|x\right|}\right]}-5}}{3+a^{\frac{1}{\left|x\right|}}}$

$=0$

$f\left(0\right)=0$

Hence, $f\left(x\right)$ is continous at $x=o$

$f^{{}^{\prime }}\left(x\right)={lim}_{h\to 0}\frac{f(x+h)-f\left(x\right)}{h}$

$⟹f^{{}^{\prime }}\left(0\right)={lim}_{h\to 0}\frac{f\left(h\right)-f\left(0\right)}{h}={lim}_{h\to 0}\frac{f\left(x\right)}{h}$

$⟹f^{{}^{\prime }}\left(0\right)=\left({\log }_{a}a\right).\frac{a^{\frac{2}{\left[\frac{-1}{\left|x\right|}\right]}-5}}{3+a^{\frac{1}{\left|x\right|}}}=\frac{a^{-5}}{3}$

Hence, $f$ is continous but not differentiable at $x=0$