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Multiple equality conditions

Prove that if \(a, b, c \) are non-negative real numbers such that \( a + b + c = 3 \), then we have

\[ ab^2 + bc^2 + ca^2 + abc \leq 4 . \]

Note by Calvin Lin
1 year, 11 months ago

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This is quite a famous inequality, and there are several approaches that could be taken. Here is the simplest proof that I know of.

Define \( f( a, b, c) = ab^2 + bc^2 + ca^2 + abc \). WLOG, we may assume that \( a \) is the median of the 3, IE we either have \( b \leq a \leq c \) or \( b \geq a \geq c \).

Step 1: We will show that \( f(a,b,c) \leq f ( a, b+c, 0 ) \). This approach is known as Smoothing.

This follows by expanding both sides, and we want to compare \( ab^2 + bc^2 + ca^2 + abc \leq a(b+c)^2 \), which simplifies to

\[ bc^2 + ca^2 \leq abc + ac^2 \]

This is equivalent to

\[ c ( a - b ) ( a - c) \leq 0 \]

From the assumption that \(a\) is the middle number, we know that \( a-b, a-c \) will have different signs (or be 0). Since \( c \) is non-negative, hence the entire product will be \( \leq 0 \).

This is essentially how we approach @Krishna Sharma 's Case 1.

Step 2: We show that subject to \( a + b = 3 \), we have \( f(a, b, 0) \leq 4 = f ( 1, 2, 0) \).

By AM-GM, we get

\[ f(a, b, 0) = ab^2 = 4 \times a \times \frac{b}{2} \times \frac{b}{2} \leq 4 \left( \frac{a + \frac{b}{2} + \frac{b}{2} } { 3} \right) ^3 = 4 .\]

Hence, the result follows.


However, I do not know of an easy way to motivate the approach, and in particular step 1. Any thoughts or comments? Calvin Lin Staff · 1 year, 11 months ago

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@Calvin Lin Sir, What is meant by IE ? Priyanshu Mishra · 1 year, 4 months ago

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@Priyanshu Mishra It's just another way to write i.e. which means "that is". Prasun Biswas · 1 year, 4 months ago

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@Calvin Lin Minor typo in the last step where you use AM-GM.

\[4\times a\times \left(\frac{b}{2}\right)^2\leq 4\left(\frac{a+\frac{b}{2}+\frac{b}{2}}{\color{red}{3}}\right)^3=4\] Prasun Biswas · 1 year, 11 months ago

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@Prasun Biswas Thanks fixed. Calvin Lin Staff · 1 year, 11 months ago

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This is just a comment, not a solution.

I remember seeing a stronger version of this inequality somewhere recently (probably on Math SE) which stated the following:

If \(a,b,c\) are non-negative reals, then the following inequality holds:

\[~\\~ab^2+bc^2+ca^2+abc\leq \frac{4}{27}(a+b+c)^3\]

Our required inequality trivially follows from the stated inequality. However, proving the said stronger inequality seems to be harder than I expected. Let me see if I can think of a proof to that. For the time being, others are welcome to post their proof (if any) for the stated stronger inequality. Prasun Biswas · 1 year, 11 months ago

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@Prasun Biswas Note that the inequalities are equivalent. What happened was that we normalized the inequality, meaning that we made all of the terms have the same polynomial degree. To do so, we multiplied, where necessary, by \( a + b + c = 3 \).

This is a standard approach, and one that I would often (though not always) recommend to use if the terms have different degrees.


The weaker version that you saw, was most likely

\[ ab^2 + bc^2 + ca^2 \leq 4 \Leftrightarrow ab^2 + bc^2 + ca^2 \leq \frac{4}{27} ( a + b + c ) ^ 3 . \] Calvin Lin Staff · 1 year, 11 months ago

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Let \[f(a,b,c)=ab^2+bc^2+ca^2+abc + k(a+b+c-3)\] (say this as equation \((1)\)

Solving these four equations \[\frac{\partial f}{\partial a} = 0\] \[\frac{\partial f}{\partial b} = 0\] \[\frac{\partial f}{\partial c} = 0\] \[a+b+c=3\]

we get \[a=b=c=1 , k=-4\]

substituting this back in equation \(1\) , we get \[f_{max}(a,b,c)=1+1+1+1-4(0) = 4\]

\(\textbf{Q.E.D}\)

@Calvin Lin sir Aman Rajput · 1 year, 6 months ago

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@Aman Rajput This demonstrates the point of this problem.

You did not perform the Lagrangian properly. At the IMO, this solution will be scored 0/7.
1. You did not state the equations.
2. You did not state how to solve the equations.
3. You missed out the equality case of \( (1, 2, 0) \). Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin I didn't miss the case 1,2,0 doesnt satisfy the third equation ,i.e. , df/dc = 0 Aman Rajput · 1 year, 6 months ago

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@Aman Rajput You missed it because you didn't take care of the boundary condition. IE You didn't perform the Lagrangian properly.

E.g. What is the maximum of \( f(x) = x^2 \) on the interval \( [-2, 2] \)? Do you say that " \(f' = 0 \Rightarrow x = 0 \) hence the maximum is \( f(0) = 0 \)? No, we still have to check the boundary points, where they need not satisfy \( f' = 0 \) in order to be a maximum on the restricted domain. Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin I know what you are trying to say . But what i know is that the min / max will be obtained using lagrangian even if we missed out boundary condition or other ordered pairs of equality Aman Rajput · 1 year, 6 months ago

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@Aman Rajput No it will not.

For example, if you ignore the boundary condition when calculating the max of \( f( x) = x^2 \) on the closed interval \( [1, -1] \), then you would not get any answer. Calvin Lin Staff · 1 year, 6 months ago

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@Calvin Lin Okay ,, ya agree ! in this question we have to check at boundary.

ok i will keep in mind always to check at boundary Aman Rajput · 1 year, 6 months ago

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@Aman Rajput This problem becomes quite a standard exercise when you use Lagrange multipliers. I think that the challenge is to prove the given inequality without using that which would be the reason for the problem being tagged under Algebra. Prasun Biswas · 1 year, 6 months ago

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@Prasun Biswas i dont think so ... :) Aman Rajput · 1 year, 6 months ago

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@Aman Rajput What Prasun said is actually true. Lagrange Multiplier makes this problem too easy. It's like you're using a chainsaw to cut a carrot.

On the other hand, you need to show that the extremal point you've found is a global maximum point as opposed to an inflection point or a global minimum point. Pi Han Goh · 1 year, 6 months ago

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Comment deleted Apr 22, 2015

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@Krishna Sharma Case 1 doesn't work. When you calculated \( \frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq 3 \), you found the minimum of the expression, but we want the maximum of the expression.
Also, you cannot "Say we want to maximize \( fg \), then we maximize \(g\) and see what happens". This is what you are doing by saying "the maximum of \(abc\) occurs at \(a = b = c = 1 \)". In particular, when \( a = 2, b = 0.1 \), note that the expression \( f \geq 20 \), and so you have to explain how to compensate for that.

For case 2, your equality case is \( a + b = 3, a = \frac{b}{2} \), yielding \( (1,2,0) \). I'm not sure why you listed \( (2,1,0 ) \). Calvin Lin Staff · 1 year, 11 months ago

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Comment deleted Apr 20, 2015

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@Pi Han Goh Saying "apply Lagrange Multiplier" is a cop-out, unless you actually work through the details.

In the IMO, the above proof would be worth 0. Calvin Lin Staff · 1 year, 11 months ago

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Comment deleted Apr 20, 2015

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@Pi Han Goh It is not trivial to show that "..." obtain their maximium value at \( a = b = c = 1 \). In fact, none of them do. For example, take \( a = b = 1.5 \) in any of these, and we obtain a higher value.

Note: Given that the equality conditions are \( (1,1,1), (1,2,0), (0,1,2), (2,0,1) \), it is hard for any of the "classical inequalities" to be applied in a direct manner. Calvin Lin Staff · 1 year, 11 months ago

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Is some info missing? Because substituting \(a=b=c = \frac{4}{3}\) yields a value around 9.5 Krishna Sharma · 1 year, 11 months ago

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@Krishna Sharma Ooops, the condition should have been \( a+b+ c = 3 \). Calvin Lin Staff · 1 year, 11 months ago

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Why not assume a=b= c=1 and do it Shailesh Hegde · 1 year, 11 months ago

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