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

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

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

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

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TopNewestThis 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, 6 months ago

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– Priyanshu Mishra · 11 months, 3 weeks ago

Sir, What is meant by IE ?Log in to reply

i.e. which means "that is". – Prasun Biswas · 11 months, 3 weeks ago

It's just another way to writeLog in to reply

\[4\times a\times \left(\frac{b}{2}\right)^2\leq 4\left(\frac{a+\frac{b}{2}+\frac{b}{2}}{\color{red}{3}}\right)^3=4\] – Prasun Biswas · 1 year, 6 months ago

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– Calvin Lin Staff · 1 year, 6 months ago

Thanks fixed.Log in to reply

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:

Our required inequality trivially follows from the stated inequality. However, proving the said

strongerinequality 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 statedstrongerinequality. – Prasun Biswas · 1 year, 6 months agoLog in to reply

normalizedthe 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, 6 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, 1 month ago

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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, 1 month ago

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– Aman Rajput · 1 year, 1 month ago

I didn't miss the case 1,2,0 doesnt satisfy the third equation ,i.e. , df/dc = 0Log in to reply

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, 1 month ago

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– Aman Rajput · 1 year, 1 month ago

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 equalityLog in to reply

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, 1 month ago

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

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– Prasun Biswas · 1 year, 1 month ago

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.Log in to reply

– Aman Rajput · 1 year, 1 month ago

i dont think so ... :)Log in to reply

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, 1 month ago

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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, 6 months ago

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In the IMO, the above proof would be worth 0. – Calvin Lin Staff · 1 year, 6 months ago

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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, 6 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, 6 months ago

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– Calvin Lin Staff · 1 year, 6 months ago

Ooops, the condition should have been \( a+b+ c = 3 \).Log in to reply

Why not assume a=b= c=1 and do it – Shailesh Hegde · 1 year, 6 months ago

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Myth: An expression attains it's maximum or minimum when all (some) of the variables are equal. – Pi Han Goh · 1 year, 6 months ago

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