If x,y,z∈R, x+y+z=5, x2+y2+z2=9𝑥,𝑦,𝑧∈𝑅, 𝑥+𝑦+𝑧=5, 𝑥2+𝑦2+𝑧2=9, then length of interval in which x𝑥 lies is

To find the length of the interval in which x lies, we can use the given equations and the Cauchy-Schwarz inequality.

The Cauchy-Schwarz inequality states that for any real numbers a1, a2, ..., an and b1, b2, ..., bn, the following inequality holds:

(a1b1 + a2b2 + ... + an*bn)^2 ≤ (a1^2 + a2^2 + ... + an^2) * (b1^2 + b2^2 + ... + bn^2)

In this case, we can set a1 = a2 = a3 = 1 and b1 = x, b2 = y, b3 = z. Then the inequality becomes:

(x + y + z)^2 ≤ 3 * (x^2 + y^2 + z^2)

Substituting the given equations, we get:

5^2 ≤ 3 * 9

25 ≤ 27

Since the inequality holds, we know that the values of x, y, and z are possible.

To find the interval in which x lies, we can use the fact that x = 5 - y - z and substitute this into the second equation:

(5 - y - z)^2 + y^2 + z^2 = 9

Solving this equation for x gives the possible values of x.

However, without additional information or constraints on the values of y and z, we cannot determine a specific interval for x. The problem may be underdetermined.