Question

Let $f\left(x\right)=x^3$ and $g\left(x\right)=\left|x\right|$. Then at $x=0$, the composite functions-

• A. gof is derivable but fog is not
• B. fog is derivable but gof is not
• C. gof and fog both are derivable
• D. neither gof nor fog is derivable

Answer 1

The correct option is C gof and fog both are derivable

Given the two functions, $gof=|x^3|$ and $fog=|x{|}^3$

Now, let $p\left(x\right)=|x^3|$ and $q\left(x\right)=|x{|}^3$

$p^{\prime }\left(0^{-}\right)=li{m}_{h\to 0}\frac{|h^3|}{-h}=-h^2=0$

$p^{\prime }\left(0^{+}\right)=li{m}_{h\to 0}\frac{|h^3|}{h}=h^2=0$

$q^{\prime }\left(0^{-}\right)=li{m}_{h\to 0}\frac{|h{|}^3}{-h}=h^2=0$

$q^{\prime }\left(0^{+}\right)=li{m}_{h\to 0}\frac{|h{|}^3}{h}=h^2=0$

Since $q^{\prime }\left(0^{-}\right)=q^{\prime }\left(0^{+}\right)$ and $p^{\prime }\left(0^{-}\right)=p^{\prime }\left(0^{+}\right)$, both the composite functions are differentiable.