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

Gujarat BoardEnglish MediumSTD 12 ScienceMaths5. continuity and differentiation1 Mark
MCQ
Let $\mathrm{f}: R \rightarrow R$ be function defined as

$f ( x )=\left\{\begin{array}{cc}3\left(1-\frac{| x |}{2}\right) & \text { if }| x | \leq 2 \text { } \\ 0 & \text { if }| x |>2 \text { }\end{array}\right.$ Let $g: R \rightarrow R$ be given by $g(x)=f(x+2)-f(x-2)$. If $n$ and $m$ denote the number of points in $R$ where $\mathrm{g}$ is not continuous and not differentiable, respectively, then $\mathrm{n}+\mathrm{m}$ is equal to $....$

  • ✓
    $4$
  • B
    $3$
  • C
    $2$
  • D
    $1$
✓

Answer

Correct option: A.
$4$
a
$f(x-2)\rightarrow =\frac{3 x}{2} \quad\quad -4 \leq x \leq-2$

$\quad \quad \quad \quad \quad \quad -\frac{3 x}{2} \quad -2\,$

$\quad \quad \quad \quad \quad \quad 0 \quad \quad \quad x \in(-\infty,-4) \cup(0,+\infty)$

$f(x+2)\rightarrow\frac{3 x}{2}+6 \quad \quad 0 \leq x \leq 2$

$\quad \quad \quad \quad \quad -\frac{3 x}{2}+6 \quad 2\,<\,x \leq 4$

$\quad \quad \quad \quad \quad 0 \quad \quad \quad \quad x \in(-\infty, 0) \cup(4,+\infty)$

$g(x)=f(x+2)-f(x-2) \rightarrow\frac{3 x}{2}+6 \quad -4 \leq x \leq-2$

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad -\frac{3 x}{2} \quad \quad -2\, < \,x\, < \,2$

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \frac{3 x}{2}-6\quad \quad 2 \leq x \leq 4$

$\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad 0 \quad \quad \quad \quad x \in(-\infty,-4) \cup(4,+\infty)$

$n=0$

$m=4 \Rightarrow(n+m=4)$

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