# Inequality for nth power of x

#### Seppel

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

#### Idea

Let $$\displaystyle f(x)=x^n-a x-b$$

show that $$\displaystyle f(\sqrt[n-1]{2a} + \sqrt[n]{2b})>0$$

Graph

#### Attachments

• 5.7 KB Views: 2
Last edited: