Problem 355 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \frac{a^{3}+b^{3}+c^{3}}{3}>\left(\frac{a+b+c}{3}\right) \cdot\left(\frac{a^{2}+b^{2}+c^{2}}{3}\right) $$

Step-by-Step Solution

Answer

The inequality is valid. The rearrangement of terms allows us to apply the triangle inequality, thereby proving the inequality is correct for all \(a\), \(b\), and \(c\)

Step 1: Clarification

This problem is an inequality. We want to prove that the inequality is valid for all \(a\), \(b\), and \(c\). In other words, we need to show that the cube of the averages is always less than or equal to the product of the average of the numbers with their squares.

Step 2: Rearrangement

Rewrite the inequality in a way that it combines like terms. Note, we are dealing with cubic and square terms. A common technique when facing such problems is to rewrite them in a way that combines like terms. So let's write \(x=a-b\), \(y=b-c\), and \(z=c-a\). The inequality can be rewritten as:\[\frac{x^2+y^2+z^2}{3} ≥ \frac{(x+y)^2+(y+z)^2+(z+x)^2}{9}\]

Step 3: Simplification

The inequality can be simplified to:\[x^2+y^2+z^2 ≥ 2(xy+yz+zx)\]This is true by the triangle inequality in geometry, which states that the square of one side of a triangle is always less than or equal to the sum of the squares of the other two sides.

Key Concepts

Algebraic Inequalities

Algebraic inequalities are fundamental components of mathematics that represent the relationship between expressions that are not equal. In essence, they determine the relative magnitude of two quantities. When solving inequalities, the goal is often to find the set of values that makes the inequality true.

It's important to be aware that operations affecting the entire inequality — such as multiplying or dividing by a negative number — can reverse the inequality sign. In the case of our exercise, we study a more complex inequality involving cubic and square terms. The provided solution illustrates how algebraic properties and geometrical concepts, like the triangle inequality, can be used in tandem to prove such inequalities.

Cubic and Square Terms

In algebra, cubic terms are those raised to the third power, and square terms are raised to the second power. They are the building blocks of polynomial expressions and are especially significant in equations and inequalities.

Understanding the relationship between cubic and square terms often requires manipulating the given expressions to reveal underlying patterns or simplify the inequality. As shown in the solution, by setting up the right substitutions and rearrangements, the originally complex expression can be rewritten to show the link between these terms in a clear and manageable form.

Triangle Inequality

The triangle inequality is a principle from geometry, but it also plays a critical role in algebraic contexts. It states that in any triangle, the length of a side is always less than the sum, and more than the difference, of the lengths of the other two sides. Mathematically, for any triangle with sides of lengths 'a,' 'b,' and 'c,' we have:
\[ a < b + c \] and \[|b - c| < a \] In the context of our inequality proof, the triangle inequality is used to assert that the sum of the squares of any two sides of a triangle is greater than the square of the third side. This helps us establish the inequality \[x^2+y^2+z^2 \geq 2(xy+yz+zx)\], paving the way for a mathematical simplification.

Mathematical Simplification

Mathematical simplification involves reducing complexity in mathematical expressions to make them easier to manage, interpret or solve. Simplification can involve factoring, combining like terms, or applying mathematical properties and theorems such as distributive, associative, and commutative laws.

In the context of our inequality, the substitution step that replaced 'a,' 'b,' and 'c' with 'x=a-b,' 'y=b-c,' and 'z=c-a' transformed the inequality into a more manageable form, enabling the use of the triangle inequality. This kind of simplification is crucial in mathematical proofs and solving problems because it makes the complex relationship between terms clear and often reveals the path to the solution.