The function $f(x) = \left\{ \begin{array}{r} |x - 3|,\ \ \ \ \ \ \ \ \ if\ x \geq 1 \ \frac{x^{2}}{4} - \frac{3x}{2} + \frac{13}{4},\ \ \ \ if\ x

Question

The function $f(x) = \left\{ \begin{array}{r} |x - 3|,\ \ \ \ \ \ \ \ \ if\ x \geq 1 \ \frac{x^{2}}{4} - \frac{3x}{2} + \frac{13}{4},\ \ \ \ if\ x < 1 \end{array} ight.\$ is

Dr. Abubacker · Accepted Answer

(b)
Clearly, $f(x)$ is not differentiable at $x = 3$
Now, $\lim_{h ightarrow 3^{-}}\ f(x) = \lim_{h ightarrow 0}\ f(3 - h)$
$= \lim_{h ightarrow 0}\ \ |3 - h - 3|$
$= 0$
$\lim_{h ightarrow 3^{+}}\ f(x) = \lim_{h ightarrow 0}\ f(3 + h)$
$= \lim_{h ightarrow 0}\ \ |3 + h - 3| = 0$
and $f(3) = |3 - 3| = 0$
$	herefore\ \ f(x)$ is continuous at $x = 3$

Q. The function $f(x) = \left\{ \begin{array}{r} |x - 3|,\ \ \ \ \ \ \ \ \ if\ x \geq 1 \\ \frac{x^{2}}{4} - \frac{3x}{2} + \frac{13}{4},\ \ \ \ if\ x < 1 \end{array} \right.\$ is

Explanation

Reviewed by Dr. Abubacker

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