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Given non-negative real numbers $a, b, c$, show that $$2a + 2ab + abc \leq 18$$ when $a + b + c = 5$

I have already started with AM-GM inequality, but now I am stuck.

From AM-GM inequality, I found that

$$abc \leq \frac{125}{27}$$

but now I am not sure what I have other two terms to finish the proof.

Maybe I am going in the wrong direction. Please help and thank you. It will be great if you can show a step-by-step solution.

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    $\begingroup$ It's often helpful to find the equality condition, which then suggests what to try (and what not to try). In this case, equality occurs at $ a = 3, b = 2, c = 0 $, so knowing that $ abc \leq 125/27$ when $ a = b = c = 5/3$ will likely not be that helpful. $\endgroup$
    – Calvin Lin
    Jun 30, 2021 at 16:43
  • $\begingroup$ Try adapting my approach to this question. Given the very similar setup, I'm guessing that this could work out. $\endgroup$
    – Calvin Lin
    Jun 30, 2021 at 16:48

1 Answer 1

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The idea is to reduce the number of variables, adapting from this other solution.

  1. Suppose that $b+c = k$, how can we maximize $2b + bc $? What is the maximum value in terms of $k$?

$ 2b + bc = b (2+c) = b ( 2 + k - b ) \leq ( \frac{2+k}{2} ) ^2$.
The maximum occurs when $ b = 2 + c = \frac{k+2}{2}$, with maximum value $ \frac{1}{4} ( k+2)^2$.

  1. Hence, for fixed $a$, what is the maximum of $ a(2+2b+bc)$ subject to $b+c = 5 - a$?

It is $ a ( 2 + \frac{1}{4} ( 7-a)^2)$.

  1. For $ a \in [ 0, 5 ]$, what is the maximum of the above cubic?

Using calculus, the maximum occurs when $a = 3$ (with $ b = 2, c = 0$) and has value 18.
If you don't want a calculus approach, see the link for how to maximize a cubic.

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  • $\begingroup$ I notice how you can use the non-calculus approach to maximize a cubic; however, the approach does not seem to work when we have a cubic polynomial like $Y = x^3 + 3x$. Is there a certain rule to say we can use $Y - D = P - D$, where $P = ax^3 + bx^2 + cx + d$? $\endgroup$
    – Jason Chiu
    Jul 1, 2021 at 5:03
  • $\begingroup$ @JasonChiu In the question you posted, Trefor's comment about checking monotonicity via the quadratic is a good approach. $\endgroup$
    – Calvin Lin
    Jul 1, 2021 at 18:09

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