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Printjmc
algebra senior
Problem
Let and be nonnegative real numbers such that Find the maximum value of
Solution
Our strategy is to add a number of inequalities like so that when we add them up, we get an inequality of the form To do so, we will use some variables, to make sure we use the most general forms of AM-GM.
If we apply AM-GM to two terms, one of which is then to obtain on the right-hand side, the other term must be as in Note that equality holds when or Thus,
We then want an inequality of the form where and are coefficients that we want to fill in. We want equality to hold here for the same values of and as in . This means we want or So, let and : Finally, should be so that we obtain on the right-hand side: Thus, we have the inequalities When we add these up, we want the coefficients of and to be equal. Thus, Isolating in we find Then Cross-multiplying, we get Substituting we get Then This simplifies to Fortunately, this polynomial has as a root.
Then and we get Therefore, Equality occurs when and so the maximum value is
If we apply AM-GM to two terms, one of which is then to obtain on the right-hand side, the other term must be as in Note that equality holds when or Thus,
We then want an inequality of the form where and are coefficients that we want to fill in. We want equality to hold here for the same values of and as in . This means we want or So, let and : Finally, should be so that we obtain on the right-hand side: Thus, we have the inequalities When we add these up, we want the coefficients of and to be equal. Thus, Isolating in we find Then Cross-multiplying, we get Substituting we get Then This simplifies to Fortunately, this polynomial has as a root.
Then and we get Therefore, Equality occurs when and so the maximum value is
Final answer
\frac{4}{3}