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?
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1$\begingroup$ 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. $\endgroup$– AxyuSJun 20, 2020 at 5:31
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1$\begingroup$ @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. $\endgroup$– ImBatmanJun 20, 2020 at 17:14
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$\begingroup$ Yesterday only I read somewhere that sum can be min at equal terms if their product is a constant. Can somebody prove that? $\endgroup$– Aman KumarJun 21, 2020 at 6:38
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1$\begingroup$ That is a direct result of the AM-GM inequality. Think about it for a moment. $\endgroup$– SamJun 24, 2020 at 13:36
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It seems that AM-GM inequality is not of much use here. Your best bet is probably to differentiate the equation and then check for the global minima by differentiating again.
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1$\begingroup$ Differentiation is often a tedious process. I am interested in knowing whether can I always say that the minimum value occurs when both of the terms are equal. $\endgroup$ Jun 20, 2020 at 5:33
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$\begingroup$ The minimum will most likely never occur when the terms are equal. $\endgroup$– SamJun 20, 2020 at 5:39