Question

The value of the derivative of $\left|\text{x-1}\right|+\left|\text{x-3}\right|$ at $\text{x=2}$ is

Answer 1

Modulus Function : It is a function which gives the absolute value of a number or variable . It gives the absolute value of variable irrespective of its sign. This function is also called the absolute value function .

$\left|x\right|=\left\{\text{x, when x≥0}\right\}and\left|\text{x}\right|=\left\{\text{-x,when x<0}\right\}$

Now we have the function $\text{f(x)=}\left|\text{x-1}\right|\text{+}\left|\text{x-3}\right|$and

At $\text{x=2}$

$\left|\text{x-1}\right|=\text{x-1 and}\left|\text{x-3}\right|\text{=-(x-3)}$

Now we define the function at $\text{x=2}$, as

$\text{f(x)=(x-1)+(-(x-3)=x-1-x+3=2}$

$\text{⇒f(x)=2 at x=2}$

So the function is a constant function .

Constant function : A function whose range consists of a single element is called the constant function such that $\text{f(x)=a}$ for any value of $x$.

We know that derivative of a constant function is always zero .So $\frac{\text{d}}{\text{dx}}\left(2\right)=0$

Hence the derivative of the given function is zero .