The function defined by f(x) = l |x - 3| ,; , , , , , , , , , , ,… - Vidyadip

MCQ

The function defined by $f(x) = \left\{ \begin{array}{l}|x - 3|\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \ge 1\\\frac{1}{4}{x^2} - \frac{3}{2}x + \frac{{13}}{4};\,x < 1\end{array} \right.$ is

A
Continuous at $x = 1$
B
Continuous at $x = 3$
C
Differentiable at $x = 1$
D
All the above

✓

Answer

Since $|x - 3|\, = x - 3,$ if $x \ge 3$
$ = - x + 3,$ if $x < 3$
$\therefore $ The given function can be defined as
$f(x) = \left\{ {\begin{array}{*{20}{r}}{\frac{1}{4}{x^2} - \frac{3}{2}x + \frac{{13}}{4},}&{x < 1\,\,\,\,\,\,\,\,}\\{3 - x,}&{1 \le x < 3}\\{x - 3,}&{x \ge 3\,\,\,\,\,\,\,}\end{array}} \right.$
Now proceed to check the continuity and differentiability at $x = 1.$

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