# inequality proving problem

#### hearts123

##### New member
Hi everyone! I have a proving problem that I don't know how to do, at all. Any help is greatly appreciated!!

Given a and b as positive real numbers, and a is not equal to b
Prove that a^5 + b^5 > (a^3)(b^2) + (a^2)(b^3)

#### pka

##### Elite Member
Given a and b as positive real numbers, and a is not equal to b
Prove that $$\displaystyle a^5 + b^5 > (a^3)(b^2) + (a^2)(b^3)$$
\displaystyle \begin{align*}(a+b)^5 &=a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5\\ &\ge5a^4b+10a^3b^2+10a^2b^3+5ab^4 \\&\ge 10a^3b^2+10a^2b^3\\& \text{Can you finish?} \end{align*}