Question

How do I find the lower bound of a function?

Answer 1

Hint: In this question, we have to find the lower bound of a function.
First we should get rid of infinities and then check the derivative.
If you divide a polynomial function $f\left( x \right)$ by $\left( {x - c} \right)$, where $c < 0,$ using synthetic division and this yields alternating signs, then $c$ is a lower bound to the real roots of the equation $f\left( x \right) = 0.$
Finally we can get the required lower bound of a function.

Complete step-by-step solution:
First of all, let’s get rid of infinities: a function can tend to $ \pm \infty $ either at the extreme points of its domain or because of some vertical asymptote.
So, we can check if the function is lower bound or not.
That is we can write it as, $\mathop {\lim }\limits_{x \to {x_0}} f\left( x \right)$
Now, we know that for every point ${x_0}$ at the boundary of the domain
For example, if the domain is $\left( { - \infty ,\infty } \right)$ , you should check $\mathop {\lim }\limits_{x \to \pm \,\infty } f\left( x \right)$
Also, the domain is like real numbers $\Re $ we should check
$\mathop {\lim }\limits_{x \to \pm \,\infty } f\left( x \right),\mathop {\lim }\limits_{x \to \,{2^{\, \pm }}} f\left( x \right),....$
Also, if any of these limits is $ - \infty $, the function has no finite lower bound.
Otherwise, you can check the derivative: when you set $f'\left( x \right) = 0,$ you will find points of maximum or minimum.
Also, for every $x$ which solves $f'\left( x \right) = 0,$ you should compute $f''\left( x \right) = 0$
If $f''\left( x \right) > 0$, the point is indeed a minimum
Now, we have to form the general case, you have a collection of points ${x_1}\,,...,{x_n}$ such that
$f'\left( {{x_i}} \right) = 0,$$f''\left( {{x_i}} \right) > 0$ For every $i = 1,...,n$
This means that they are all local minima of your function. The lower bound of the function, that is the global minimum, will be the smallest image of those points: you just need to compare
$f\left( {{x_1}} \right)\,,...,f\left( {{x_n}} \right).$ and choose the smallest one.

Note: A function $f$ is said to have a lower bound $c$ if $c \leqslant f\left( x \right)$ for all $x$ in its domain. The greatest lower bound is called the infimum.
Special note that zeros can be either positive or negative.
Note that two things must occur for $c$ to be a lower bound.
A set with an upper bound is said to be bounded from above.
To find the lower bound, subtract the upper bound of $X - Y$ from the lower bound of $X.$