Question

Test the continuity and differentiability of the function defined as under at x=1 and x=2.$f\left(x\right)=\{\begin{array}{cc}x,& x<1\\ 2-x,& 1\le x\le 2\\ -2+3x-x^2& x>2\end{array}$

• A. Continuous and differentiable at both points
• B. Dis-Continuous and not -differentiable at both points
• C. Continuous at both points but differentiable at $x=2$ only
• D. Continuous at both points but differentiable at $x=1$ only

Answer 1

The correct option is C Continuous at both points but differentiable at $x=2$ only

$f\left(x\right)=\{\begin{array}{cc}x,& x<1\\ 2-x,& 1\le x\le 2\\ -2+3x-x^2& x>2\end{array}$
$f(1-)=1$ and $f(1+)=1$
$f$ is continuous at $x=1$
$f(2-)=0$ and $f(2+)=0$
$f$ is continuous at $x=2$
$f^{\prime }\left(x\right)=\{\begin{array}{cc}1,& x<1\\ -1& 1\le x\le 2\\ 3-2x& x>2\end{array}$
$f^{\prime }(1-)=1$ and $f^{\prime }(1+)=-1$
$f$ is not differential at $x=1$
$f^{\prime }(2-)=-1$ and $f^{\prime }(2+)=-1$
$f$ is differential at $x=2$