Inequality for nth power of x

Nov 2009
11
6
Hello!

I have to solve the following exercise.

Let \(\displaystyle a\) and \(\displaystyle b\) be positive real numbers and \(\displaystyle n\) a natural number with \(\displaystyle n \geq 2\). Prove the following statement:

For all positive real numbers \(\displaystyle x\) that satisfy the inequality \(\displaystyle x^n \leq ax + b\), also \(\displaystyle x < \sqrt[n-1]{2a} + \sqrt[n]{2b}\) holds.

I am trying a proof by induction and was able to prove the base case n=2, but I don't know how to do the induction step.

Could you give me a hint? Thanks!

Seppel
 
Jun 2013
1,096
573
Lebanon
Let \(\displaystyle f(x)=x^n-a x-b\)

show that \(\displaystyle f(\sqrt[n-1]{2a} + \sqrt[n]{2b})>0\)
 
Jun 2013
1,096
573
Lebanon
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