I am preparing for my exam and need help with the following task:
Let be a continuous function with the estimation . Show that f takes on its absolute minima.
If a function is continuous, then =.
Our function has an absolute minima in , if for .
At first I thought the task is pretty easy. We learned how to prove that if 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 . This gives us the information, that f is bounded from below with . But how does this help me? And what else do we have?