Prove $X\ge 0$ and Find Smallest Positive $X$ | POTW #158 April 8, 2015

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In summary, to prove that X is greater than or equal to 0, we can use mathematical induction or direct proof. It is important to prove this because it helps us understand the properties and behavior of X, which is often used to represent physical quantities in equations. Finding the smallest positive value of X is significant as it shows the lower bound and helps in making comparisons and solving optimization problems. While X can be equal to 0, proving that it is greater than or equal to 0 for all other values is necessary to show that it holds for all possible cases. This concept is also relevant in real-life situations, such as in physics, economics, and everyday scenarios where certain values cannot be negative.
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Let $a,\,b$ and $c$ be integers, and given $X=16(a^2+b^2+c^2)-5(a+b+c)^2$, prove that $X\ge 0$ and find the smallest positive number that $X$ can be.

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Congratulations to the following members for their correct solutions::)

1. kaliprasad
2. greg1313

Solution from kaliprasad:

$\begin{align*}X&= 16(a^2+b^2+c^2) - 5(a+b+c)^2\\&=16(a^2+b^2+c^2) - 5(a^2+b^2+c^2+2ab+2ac+2bc)\\&=11(a^2+b^2+c^2) - 10(ab+bc+ca)\\&=a^2+b^2+c^2 + 5(2a^2+2b^2+2c^2-2ab - 2bc - 2ca)\\&=(a^2+b^2+c^2 + 5((a-b)^2+(b-c)^2 + (c-a)^2)\end{align*}$

Clearly the lowest value is zero when $a = b= c = 0$

If $a,\,b,\,c$ are not same then the lowest value is 5 and if same and 1 or -1 then the value is 3.

So lowest positive value = 3 at $a=b= c = 1$ or $a=b=c = -1$.
 

FAQ: Prove $X\ge 0$ and Find Smallest Positive $X$ | POTW #158 April 8, 2015

How do you prove that X is greater than or equal to 0?

To prove that X is greater than or equal to 0, we can use mathematical induction or direct proof. In mathematical induction, we first show that X is greater than or equal to 0 for the base case, usually when X = 0. Then, we assume that X is greater than or equal to 0 for some arbitrary value k, and prove that it holds for k+1. This shows that X is greater than or equal to 0 for all positive integers. In direct proof, we start with the given statement that X is greater than or equal to 0, and use logical steps to show that it is true for all cases.

Why is it important to prove that X is greater than or equal to 0?

It is important to prove that X is greater than or equal to 0 because it helps us understand the properties and behavior of X. In many mathematical and scientific equations, X represents a physical quantity such as time, distance, or temperature. Proving that X is greater than or equal to 0 ensures that the value of X is always positive, which is necessary for the validity of many equations and calculations.

What is the significance of finding the smallest positive value of X?

Finding the smallest positive value of X is significant because it allows us to understand the lower bound of X. In other words, it shows us the smallest possible value that X can take, which can help us in making comparisons, estimations, and predictions. Additionally, finding the smallest positive value of X can also help us in solving optimization problems where we want to minimize X.

Can X be equal to 0?

Yes, X can be equal to 0. In fact, 0 is the smallest possible value of X. However, when we are asked to prove that X is greater than or equal to 0, it means that X must be greater than or equal to 0 for all values of X other than 0. This is because the statement "X is greater than or equal to 0" is already true for X = 0, and proving it for all other values is necessary to show that it holds for all possible cases.

How does proving that X is greater than or equal to 0 relate to real-life situations?

Proving that X is greater than or equal to 0 relates to real-life situations in many ways. For example, in physics, the position of an object can never be less than 0, as it represents a distance from a fixed point. In economics, profits are always equal to or greater than 0, as negative profits indicate losses. In everyday life, we often encounter situations where certain values cannot be negative, making it important to prove that they are greater than or equal to 0.

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