To #a): and is continuously increasing for all . So the minimum of the complete term occurs when |x| has it's minimum.

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

I don't understand the writing of the term at #a). Do you mean:Find out the maximum on |R of the following functions :

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

b- K: x--> (2/1+x²) - 3.

or

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.