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

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.

• 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. Jun 30, 2021 at 16:43
• Try adapting my approach to this question. Given the very similar setup, I'm guessing that this could work out. Jun 30, 2021 at 16:48

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.

• 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 $P = ax^3 + bx^2 + cx + d$? Jul 1, 2021 at 5:03
• @JasonChiu In the question you posted, Trefor's comment about checking monotonicity via the quadratic is a good approach. Jul 1, 2021 at 18:09