Question

The value of the derivative of $\left| {x - 1} \right| + \left| {x - 3} \right|$ at $x = 2$ is
A. $2$
B. $1$
C. $0$
D. $- 2$

Answer 1

Hint : Observe the given expression. There are moduli in the given expression. Recall the concept of modulus of a number and use this to find the terms of the given expression at $x = 2$ . Then simplify the expression and find its derivative. Keep the properties of derivatives in mind.

Complete step-by-step answer :
Given, the expression $\left| {x - 1} \right| + \left| {x - 3} \right|$
We are asked to find its derivative at $x = 2$
Let $f(x) = \left| {x - 1} \right| + \left| {x - 3} \right|$ (i)
In the function we observe two terms that is $\left| {x - 1} \right|$ and $\left| {x - 3} \right|$ are given in modulus.
Modulus of number $z$ or $\left| z \right|$ is defined as,

z&{{\text{if}}\,z \geqslant 0} \\\ { - z}&{{\text{if}}\,z \leqslant 0} \end{array}} \right\\}$$ That means the modulus of $$z$$ is positive when $$z$$ is greater than zero and is negative when $$z$$ is less than zero. Let us check for both terms of function $$f(x)$$ at $$x = 2$$ whether they are positive or negative. At $$x = 2$$ , $$\left| {x - 1} \right| = \left| {2 - 1} \right| = \left| 1 \right| > 0$$ That is the term will be positive, $$\left| {x - 1} \right| = (x - 1)$$ (ii) At $$x = 2$$ , $$\left| {x - 3} \right| = \left| {2 - 3} \right| = \left| { - 1} \right| < 0$$ That is the term will be negative, $$\left| {x - 3} \right| = - (x - 3) = - x + 3$$ (iii) Now, we find the function at $$x = 2$$ . Using equation (ii) and (iii) in equation (i), we get $$f(x) = x - 1 - x + 3 \\\ \Rightarrow f(x) = 2 $$ We get the function as constant and we know the derivative of a constant is always zero. Therefore, the derivative of the given function at $$x = 2$$ is zero. **So, the correct answer is “Option C”.** **Note** : In such types of questions where moduli are there always check those terms first involving modulus whether they are greater than zero or less than zero, then form the expression and proceed according to the question given. Always remember the derivative of a constant is zero and the reverse is also true, that is if the derivative of a function is zero then the function is constant.