# Maximization problem in multivariable calculus

Let $f(x_1,\ldots,x_n) = x_1x_2\cdot \cdot \cdot x_n$. Let $A : = \{ x \in \mathbb{R}^n : x_1 + x_2 + \cdots + x_n = n , \; \; x_i \geq 0 \; \; \forall i \}$. I want to find the global max of $f$ under the constrain given by $A$. MY approach:

Put $F(x_1,\ldots,x_n) = \sum_{i=1}^n x_i - n = 0$ is the constrain. We use lagrange multipliers:

$$\nabla f = \lambda \nabla F \implies (x_2 \cdot \cdot \cdot x_n, \ldots, x_1 \cdots x_{n-1}) = (\lambda,\lambda,\ldots,\lambda).$$ Hence we have

$$x_2\cdot \cdot \cdot x_n = x_1 x_3 \cdot \cdot \cdot x_n = \cdots = x_1\cdot \cdot \cdot x_{n-1}$$

this implies that $$x_1 = x_2 = \cdots = x_n$$

We also know $\sum x_i = n \implies n X = n \implies X = 1$ if we let $X = x_i$

So, max occurs at $(1,1,\ldots,1)$ which is $f(1,1,\ldots,1) = 1$ Is this correct? I feel im missing something. thanks for any feedback.