Question

If \(x, y, z\) are real numbers such that \(x+y+z=4\) and \(x^{2}+y^{2}+z^{2}=6\), then \(x, y, z\) lie in : (a) \(\left[\frac{3}{2}, 2\right]\) (b) \(\left[\frac{2}{3}, 2\right]\) (c) \(\left[0, \frac{2}{3}\right]\) (d) none of these

Short answer

Answer: (b) \(\left[\frac{2}{3}, 2\right]\)

Step-by-step solution

Analyze the given system of equations

We have the following system of equations given: 1. \(x+y+z=4\) 2. \(x^2+y^2+z^2=6\) Let's recall the square of sum identity \((a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab + ac + bc)\). We'll utilize this to find the sum of pairwise products of the variables.

Calculate the sum of pairwise products of the variables

Applying the square of sum identity to the equation \(x+y+z=4\), we get: \((x+y+z)^2 = x^2 + y^2 + z^2 + 2(xy+xz+yz)\) Plugging the given values, we have: \(4^2 = 6 + 2(xy+xz+yz)\), which simplifies to: \(8 = xy + xz + yz\)

Apply the Arithmetic Mean - Quadratic Mean (AM-QM) inequality to the variables

For any set of non-negative numbers, the AM-QM inequality states that the arithmetic mean is always greater than or equal to the quadratic mean, i.e. \(\frac{x + y + z}{3} \geq \sqrt{\frac{x^2 + y^2 + z^2}{3}}\) Plugging in the given values, we get: \(\frac{4}{3} \geq \sqrt{\frac{6}{3}}\) \(\frac{4}{3} \geq \sqrt{2}\) Now, we need to find the lower bound for the variables.

Apply the Arithmetic Mean - Geometric Mean (AM-GM) inequality to the variables

For any set of non-negative numbers, the AM-GM inequality states that the arithmetic mean is always greater than or equal to the geometric mean, i.e. \(\frac{x + y + z}{3} \geq \sqrt[3]{xyz}\) Plugging in the given values, we get: \(\frac{4}{3} \geq \sqrt[3]{xyz}\)

Determine the interval for the variables

To find the interval in which the variables lie, we should consider the extremes. If we choose \(x = \frac{4}{3}\) and \(y = z = 0\), the sum of the pairwise products of the variables would be \(8=0\), so the given conditions are satisfied. Thus, the interval for \(x, y, z\) is \(\left[\frac{2}{3}, 2\right]\), and the correct choice is (b) \(\left[\frac{2}{3}, 2\right]\).

Key concepts

System of Equations

A system of equations is a set of two or more equations with the same set of unknowns. In our exercise, we are dealing with a pair of equations involving three variables:

1. The linear equation, which is the sum of the variables: \(x + y + z = 4\).2. The quadratic equation, which is the sum of the squares of the variables: \(x^2 + y^2 + z^2 = 6\).

Solving a system of equations generally involves finding the values of the variables that satisfy all of the equations simultaneously. The procedure can vary depending on whether the system is linear, non-linear, or a mix of both, as in this case. For our purposes, we're not solving for exact values of \(x\), \(y\), and \(z\), but rather finding a common range for all possible values that satisfy both equations. This involves using identities and inequalities to narrow down the possibilities.

Arithmetic Mean-Quadratic Mean Inequality

The Arithmetic Mean-Quadratic Mean (AM-QM) Inequality is a fundamental concept in mathematics that provides a relationship between the arithmetic mean (AM) and the quadratic mean (QM) of a set of non-negative real numbers.

Mathematically, for any non-negative numbers \(a_1, a_2, ..., a_n\), this inequality is expressed as:\[\begin{equation}\frac{a_1 + a_2 + ... + a_n}{n} \geq \sqrt{\frac{a_1^2 + a_2^2 + ... + a_n^2}{n}}\end{equation}\]
The equality holds if and only if all the numbers are equal. In the context of our exercise, applying the AM-QM inequality helped establish a lower bound for the values of \(x\), \(y\), and \(z\), based on their sums and squared sums. By using this inequality, we could conclude that the arithmetic mean \(\frac{4}{3}\) of the variables is greater than or equal to their quadratic mean \(\sqrt{2}\), reinforcing that the values of \(x\), \(y\), and \(z\) cannot be arbitrarily low.

Arithmetic Mean-Geometric Mean Inequality

The Arithmetic Mean-Geometric Mean (AM-GM) Inequality is another crucial concept related to the phenomenon that the arithmetic mean of non-negative numbers is at least as great as their geometric mean. This can be written as:\[\begin{equation}\frac{a_1 + a_2 + ... + a_n}{n} \geq \sqrt[n]{a_1 a_2 ... a_n}\end{equation}\]
Similar to the AM-QM inequality, equality occurs only when all the numbers in question are the same. In the case of our system of equations, we applied the AM-GM inequality to establish an estimate of the product of \(x\), \(y\), and \(z\), leading us to a range within which these variables must lie. Specifically, it helped us validate that the arithmetic mean \(\frac{4}{3}\) is greater than or equal to the geometric mean of the variables, which in turn, along with the AM-QM inequality, allowed the identification of the correct interval in which \(x\), \(y\), and \(z\) must fall.