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KCET · Maths · Continuity and Differentiability

Match the following:
In the following, \([\mathrm{x}]\) denotes the greatest integer less than or equal to x .
\begin{array}{|l|l|l|l|}\hline & \text{Column - I} & & \text{Column - II} \\\hline \text{a}. & \mathrm{x}|\mathrm{x}| & \text{(i)} & \text {continuous in (-1, 1)} \\\hline \text{b.} & \sqrt{|\mathrm{x}|} & \text{(ii)} & \text{differentiable in (-1, 1)} \\\hline \text{c.} & \mathrm{x}+[\mathrm{x}] & \text{(iii)} & \text{strictly increasing in (-1, 1)} \\\hline \text{d.} & |\mathrm{x}-1|+|\mathrm{x}+1| & \text{(iv)} & \text{not differentiable at, at least one point in (-1, 1)} \\\hline\end{array}

  1. A \(a-i, b-i i, c-i v, d-i i i\)
  2. B \(a-i v, b-i i i, c-i, d-i i\)
  3. C \(a-i i, b-i v, c-i i i, d-i\)
  4. D \(\mathrm{a}-\mathrm{iii}, \mathrm{b}-\mathrm{ii}, \mathrm{c}-\mathrm{iv}, \mathrm{d}-\mathrm{i}\)
Verified Solution

Answer & Solution

Correct Answer

(C) \(a-i i, b-i v, c-i i i, d-i\)

Step-by-step Solution

Detailed explanation

\(\begin{aligned}
& \text { (a) } x|x| \\
& f(x)=\left\{\begin{array}{ll}
x^2, & x \geq 0 \\
-x^2 & x \lt 0
\end{array} \text { differentiable in }(-1,1)\right.
\end{aligned}\)

(b) \(\sqrt{|x|}=\left\{\begin{aligned} \sqrt{x}, & x \geq 0 \\ \sqrt{-x}, & x \lt 0\end{aligned} \quad\right.\) Not differentiable at \(x=0\)
(c) \(x+[x]\) strictly increasing in \((-1,1)\)
(d) \(|x-1|+|x+1|=\left\{\begin{array}{cc}-x+1-x-1 & x \lt -1 \\ -x+1+x+1 & -1 \lt x \lt 1 \\ x-1+x-1 & x\gt1\end{array}\right.\)

Continuous \((-1,1)=\left\{\begin{array}{ccc}-2 x & , & x \lt -1 \\ 2, & -1 \lt x \lt 1 \\ 2 x & , & x\gt1\end{array}\right.\)