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5. continuity and differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
If $f(x) = \left\{ \begin{array}{l}\,\,\,\,\,\,\,\,\, - {x^2},\,{\rm{when\,\, }}x \le 0\\\,\,\,\,\,5x - 4,\,{\rm{when\,\,}}0 < x \le 1\\4{x^2} - 3x,\,{\rm{when\,\, }}1 < x < 2\\\,\,\,\,\,3x + 4,{\rm{when \,\,}}x \ge 2\end{array} \right.$, then
  • A
    $f(x)$ is continuous $x = 0$
  • ✓
    $f(x)$ is continuous $x = 2$
  • C
    $f(x)$is discontinuous at$x = 1$
  • D
    None of these
✓

Answer

Correct option: B.
$f(x)$ is continuous $x = 2$
b
(b) $\mathop {\lim }\limits_{x \to 0 - } \,\,f(x) = 0$

$f(0) = 0,\,\,\mathop {\lim }\limits_{x \to 0 + } \,\,f(x) = - 4$

$f(x)$ discontinuous at $x = 0.$

and $\mathop {\lim }\limits_{x \to 1 - } \,\,f(x) = 1$ and $\mathop {\lim }\limits_{x \to 1 + } \,\,f(x) = 1,\,\,f(1) = 1$

Hence $f(x)$ is continuous at $x = 1$.

Also $\mathop {\lim }\limits_{x \to 2 - } \,\,f(x) = 4{(2)^2} - 3\,.\,2 = 10$

$f(2) = 10$ and $\mathop {\lim }\limits_{x \to 2 + } \,\,f(x) = 3(2) + 4 = 10$

Hence $f(x)$ is continuous at $x = 2.$

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