The function $\displaystyle f$ belongs to $\displaystyle \mathcal{C}^1(A)$ where $\displaystyle A$ is the open set $\displaystyle A=\prod_{i=1}^n(a_i,b_i)$ .

On the other hand, $\displaystyle g(x_1,\ldots,x_n)=x_1+\ldots+x_n-1$ also belongs to $\displaystyle \mathcal{C}^1(A)$ and $\displaystyle \textrm{rank}(\;\nabla g(x_1,\ldots,x_n)\

=1$ for all $\displaystyle x=(x_1,\ldots,x_n)\in A$ (maximun rank).

According to the multipliers Lagrange theorem if $\displaystyle f$ has a local maximum or minimum at $\displaystyle x_0\in A$ there exists $\displaystyle \lambda\in \mathbb{R}$ such that $\displaystyle x_0$ is critical point of

$\displaystyle F(x_1,\ldots,x_n)=f(x_1,\ldots,x_n)+\lambda g(x_1,\ldots,x_n)$

Verify that such a point does not exist. This means that the absolute maximum an minimum value for $\displaystyle f$ is in the boundary $\displaystyle \partial (K)$ of the compact set:

$\displaystyle K=\left(\;\prod_{i=1}^n[a_i,b_i]\;\right)\cap\{(x_1,\ldots,x_n):x_1+\ldots+x_n=1\}$

because $\displaystyle f:K\to \mathbb{R}$ is continuous.