Let $f: R \rightarrow R$ be a function defined as $f(x)=\left\{\begin{array}{ccc} 5 & , & \text { if } x…

Let $f: R \rightarrow R$ be a function defined as $f(x)=\left\{\begin{array}{ccc} 5 & , & \text { if } x \leq 1 \\ a+b x & , & \text { if } 1 < x < 3 \\ b+5 x & , & \text { if } 3 \leq x < 5 \\ 30 & , & \text { if } x \geq 5 \end{array} \mid \text { then } f\right. \text { is }$
  1. continuous if $a=5$ and $b=5$.
  2. continuous if $a=-5$ and $b=10$.
  3. not continuous for any values of $a$ and $b$.
  4. continuous is $a=0$ and $b=5$.

Solution

for continuity at $x=1, a+b=5$ for continuity at $x=3, a+3 b=b+15$ $\Rightarrow a+2 b=15$ $\because$ system of equation (i), (ii) and (iii) is inconsistent Hence, $f(x)$ is not continuous for any values of $a$ and $b$ on $R$

Asked in: MHT CET 2022 (08 Aug Shift 2)

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