# Majorant and minorant

• September 29th 2011, 10:42 AM
Narimen
Majorant and minorant
Hi , Need some help plz :
Find out the minimum on |R of the following functions :

a - f: x --> 1 + |x| + 2x².
b - g:x --> |x+1| - 4.

Find out the maximum on |R of the following functions :

a- h: x--> (1/|x|+3) +1.
b- K: x--> (2/1+x²) - 3.
• September 29th 2011, 10:50 PM
earboth
Re: Majorant and minorant
Quote:

Originally Posted by Narimen
Hi , Need some help plz :
Find out the minimum on |R of the following functions :

a - f: x --> 1 + |x| + 2x².
b - g:x --> |x+1| - 4.

To #a): $1+2x^2>0$ and is continuously increasing for all $x \in \mathbb{R}$. So the minimum of the complete term occurs when |x| has it's minimum.

T #b): Similar argumentation as described at #a).

Quote:

Find out the maximum on |R of the following functions :

a- h: x--> (1/|x|+3) +1.
b- K: x--> (2/1+x²) - 3.
I don't understand the writing of the term at #a). Do you mean:

$h(x)=\frac1{|x|+3}+1$ or $h(x)=\left(\frac1{|x|}+3\right)+1$

To #b): The maximum of the complete term occurs when the fraction has it's maximum. Since the numerator is a constant the maximum is reached if the denominator has it's minimum. Since x² is positiv or zero the minimum of the denominator is obvious.