Answer:
Maximum value: ![3* \sqrt{n}](https://tex.z-dn.net/?f=%203%2A%20%5Csqrt%7Bn%7D%20)
Minimum value: ![-3* \sqrt{n}](https://tex.z-dn.net/?f=%20-3%2A%20%5Csqrt%7Bn%7D%20)
Step-by-step explanation:
Let
, the restriction function.The Lagrange Multiplier problem states that an extreme (x1, ..., xn) of f with the constraint g(x) = 9 has to follow the following rule:
![\nabla{f}(x_1, ..., x_n) = \lambda \nabla{g} (x_1,...,x_n)](https://tex.z-dn.net/?f=%20%5Cnabla%7Bf%7D%28x_1%2C%20...%2C%20x_n%29%20%3D%20%5Clambda%20%5Cnabla%7Bg%7D%20%28x_1%2C...%2Cx_n%29%20)
for a constant
.
Note that the partial derivate of f respect to any variable is 1, and the partial derivate of g respect xi is 2xi, this means that
![1 = \lambda 2 x_1](https://tex.z-dn.net/?f=%201%20%3D%20%5Clambda%202%20x_1%20)
Thus,
![x_i = \frac{1}{2\lambda} = c](https://tex.z-dn.net/?f=%20x_i%20%3D%20%5Cfrac%7B1%7D%7B2%5Clambda%7D%20%3D%20c%20)
Where c is a constant that doesnt depend on i. In other words, there exists c such that (x1, x2, ..., xn) = (c,c, ..., c). Now, since g(x1, ..., xn) = 9, we have that n * c² = 9, or
![c = \, ^+_- \, \sqrt{\frac{9}{n} } = \, ^+_- \frac{3}{\sqrt{n}}](https://tex.z-dn.net/?f=%20c%20%3D%20%5C%2C%20%5E%2B_-%20%5C%2C%20%5Csqrt%7B%5Cfrac%7B9%7D%7Bn%7D%20%7D%20%3D%20%5C%2C%20%5E%2B_-%20%5Cfrac%7B3%7D%7B%5Csqrt%7Bn%7D%7D%20)
When c is positive, f reaches a maximum, which is ![\frac{3}{\sqrt{n}} + \frac{3}{\sqrt{n}} + \frac{3}{\sqrt{n}} + ..... + \frac{3}{\sqrt{n}} = n * \frac{3}{\sqrt{n}} = 3 * \sqrt{n}](https://tex.z-dn.net/?f=%20%20%5Cfrac%7B3%7D%7B%5Csqrt%7Bn%7D%7D%20%20%2B%20%20%5Cfrac%7B3%7D%7B%5Csqrt%7Bn%7D%7D%20%2B%20%20%5Cfrac%7B3%7D%7B%5Csqrt%7Bn%7D%7D%20%20%2B%20.....%20%2B%20%20%5Cfrac%7B3%7D%7B%5Csqrt%7Bn%7D%7D%20%20%3D%20n%20%2A%20%20%5Cfrac%7B3%7D%7B%5Csqrt%7Bn%7D%7D%20%20%3D%203%20%2A%20%5Csqrt%7Bn%7D%20)
On the other hand, when c is negative, f reaches a minimum, ![-3 * \sqrt{n}](https://tex.z-dn.net/?f=-3%20%2A%20%5Csqrt%7Bn%7D%20)