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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability1 Mark
MCQ
The function $f(x)=\left\{\begin{array}{cc}|x-3|, & x \geq 1 \\ \frac{x^2}{4}-\frac{3 x}{2}+\frac{13}{4}, & x<1\end{array}\right.$ is
  • A
    continuous at $x=1$
  • B
    differentiable at $x=1$
  • C
    continuous at $x=3$
  • ✓
    All of these
✓

Answer

Correct option: D.
All of these
$ |x-3|$ is continuous at $x=3$, but not differentiable.
$f\left(1^{-}\right)=f\left(1^{+}\right)=f(1)=2$
$f^{\prime}\left(1^{-}\right)=f^{\prime}\left(1^{+}\right)=-1=f^{\prime}(1)$
$\therefore f(x)$ is continuous and differentiable at $x=1$.

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