# The case of minimum value in AM-GM

Suppose there are $$2$$ algebraic positive real quantities, $$a(x)$$ and $$b(x)$$. I have to find the value for $$x$$ at which $$a(x)+b(x)$$ becomes minimum. Using $$\text{AM}\geq \text{GM}$$, and that equality holds when $$a(x)=b(x)$$. So can I say that the min value occurs at this condition only?

• If you do in this way you will get the value of x for which the given equality (AM>=GM) holds and it need not be minimum. You can check this with any simple example like a(x) = x^2 + and b(x) = 6 - x^2. Commented Jun 20, 2020 at 5:31
• @Professor of Stupidity What about $a(x) = x^{2}$ and $b(x) = 1+x^{2}$ those can never be equal but minimum value of $a(x)+b(x)$is 1. Commented Jun 20, 2020 at 17:14
• Yesterday only I read somewhere that sum can be min at equal terms if their product is a constant. Can somebody prove that? Commented Jun 21, 2020 at 6:38
• That is a direct result of the AM-GM inequality. Think about it for a moment.
– Sam
Commented Jun 24, 2020 at 13:36