Question

Assertion :Consider the function $f\left(x\right)=x^2-2x$ and $g\left(x\right)=-\left|x\right|$

The composite function $F\left(x\right)=f\left(g\left(x\right)\right)$ is not derivable at $x=0$

Reason: $f^{\prime }\left(0^{+}\right)=2$ and $f^{\prime }\left(0^{-}\right)=-2$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$f\left(x\right)=f\left(g\left(x\right)\right)={\left(g\left(x\right)\right)}^2-2g\left(x\right)={\left|x\right|}^2+2\left|x\right|$

$=\{\begin{array}{c}\begin{array}{cc}{x}^2+2x;& x\ge 0\end{array}\\ \begin{array}{cc}{x}^2-2x;& x<0\end{array}\end{array}$

$\Rightarrow f^{\prime }\left(x\right)=\{\begin{array}{c}\begin{array}{cc}2x+2;& x\ge 0\end{array}\\ \begin{array}{cc}2x-2;& x<0\end{array}\end{array}$

Clearly $F\left(x\right)$ is continuous at $x=0$ but $f^{\prime }\left(0^{-}\right)=-2$ and $f^{\prime }\left(0^{+}\right)=2$

$\therefore F\left(x\right)=f\left(g\left(x\right)\right)$ is non differentiable at $x=0$

Therefore assertion as well as reason both are correct and reason explain the assertion.