I am preparing for my exam and need help with the following task:Let f :...

Hayley Bernard

Hayley Bernard

Answered

2022-07-18

I am preparing for my exam and need help with the following task:
Let f : R R be a continuous function with the estimation x 2 f ( x ) x R . Show that f takes on its absolute minima.
If a function is continuous, then lim x a f ( x ) = f ( a ).
Our function has an absolute minima in x 0 R , if f ( x ) f ( x 0 ) for x R .
At first I thought the task is pretty easy. We learned how to prove that if f : [ a , b ] R is continuous, then f has an absolute maxima and an absolute Minima in [a,b]. The Problem here is, that the domain of our function here is unbounded. Thats why I don't have any idea what I could and should use for the proof. We should probably use the estimation x 2 f ( x ) x R . This gives us the information, that f is bounded from below with f ( x ) 0 x R . But how does this help me? And what else do we have?

Answer & Explanation

dasse9

dasse9

Expert

2022-07-19Added 12 answers

Let a = f ( 0 ) 0. Then, if | x | a , we have:
(1) f ( x ) x 2 a = f ( 0 )
As [ a , a ] is compact and f is continuous on it, there is some x 0 [ a , a ], such that:
f ( x 0 ) = min [ a , a ] f
Clearly, f ( x 0 ) f ( 0 ). Therefore by (1), we have:
f ( x 0 ) = min R f
jlo2ni5x

jlo2ni5x

Expert

2022-07-20Added 8 answers

It helps, thank you

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