Question

Find the local minimum value of the function f given by $f\left( x \right) = 3 + \left| x \right|,\,x \in R$.

Answer 1

Hint: First, find the minimum value of |x|. As the function is in the mod, it means it will always give a positive value. So, the minimum value of |x| will be 0. Now add 3 in the minimum value of |x|. The output 3 is the local minima of the function f.

Complete step-by-step answer:
Given: - The function $f\left( x \right) = 3 + \left| x \right|$, $x \in R$.
A modulus function is a function that gives an absolute value to a number or variable. This results in the magnitude of the number of variables. It is often referred to as the function of absolute value. The output of the function is always positive irrespective of the data.
When x is positive, it will return,
|x| = x
When x is positive, it will return,
|x| = -x
So, it can be redefining the modulus function as,
$f\left( x \right) = \left\{ \begin{gathered}
  x, & if\,x \geqslant 0 \\
   - x, & if\,x < 0 \\
\end{gathered} \right.$
It means if the value of x is greater than or equal to 0, the function will return the actual value. But if the value of x is less than 0, it will take out the negative sign and return the value.
As we know,
$\left| x \right| \geqslant 0$
Now add 3 on both sides,
$3 + \left| x \right| \geqslant 3 + 0$
Add the terms on the right side.
$f\left( x \right) \geqslant 3$

Hence, the minimum value of the function f is 3.

Note: It can also be done in another way.
The value of the function at x=0 is,
$f\left( 0 \right) = 3$
The value of the function at x=1 is,
$f\left( 1 \right) = 3 + \left| 1 \right|$
The absolute function will return 1 as its output.
$f\left( 1 \right) = 4$
The value of the function at x=-1 is,
$f\left( { - 1} \right) = 3 + \left| { - 1} \right|$
The absolute function will return 1 as its output.
$f\left( { - 1} \right) = 4$
It shows that either the value of x is increasing or decreasing, the value of f is increasing.
Hence, the local minimum value of the function is 3.