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.

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- Sep 29th 2011, 10:42 AM #1

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## 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.

- Sep 29th 2011, 10:50 PM #2
## Re: Majorant and minorant

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

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

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

a- h: x--> (1/|x|+3) +1.

b- K: x--> (2/1+x²) - 3.

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

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.