The function f(x) = [(1 - x), ,(1 + x), ,2], x ( - , , ), is - Vidyadip

MCQ

The function $f(x) = \max [(1 - x),\,(1 + x),\,2],$ $x \in ( - \infty ,\,\infty ),$ is

A
Continuous at all points
B
Differentiable at all points
✓
Differentiable at all points except at $x = 1$ and $x = - 1$
D
Continuous at all points except at $x = 1$ and $x = - 1$ where it is discontinuous

✓

Answer

Correct option: C.

Differentiable at all points except at $x = 1$ and $x = - 1$

c
(a) $f(x) = \max \,\,\left\{ {(1 - x),\,\,(1 + x),\,\,2} \right\},\,\,\forall \,\,x \in ( - \,\infty ,\,\infty ).$

$f(x) = \left\{ {\begin{array}{*{20}{r}}{1 + x;}&{x > 1}\\{2;}&{ - 1 \le x \le 1}\\{1 - x;}&{x < - 1}\end{array}} \right.$

Since $f(x) = 1 - x$ or $1 + x$ are polynomial functions and $f(x) = 2$ is a constant function.

$\therefore$ These are continuous at all points.....$(i)$

$\therefore$ $f(x)$ is differentiable at all the points, except at

$x = 1$ and at $x = - 1$.....$(ii)$

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