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## Finite value

Multivariate Calculus
Tolaso J Kos
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### Finite value

Let $\mathcal{C} =[0, 1] \times [0, 1] \times \cdots \times[0, 1] \subseteq \mathbb{R}^n$ be the unit cube. Define the function

$$f\left ( x_1, x_2, \dots, x_n \right )= \frac{x_1 x_2 \cdots x_n}{x_1^{a_1} + x_2^{a_2} + \cdots + x_n^{a_n}}$$

where $a_i$ arbitrary positive constants. For which values of $a_i>0$ is the value of the integral είναι $\bigintsss_{\mathcal{C}} \; f$ finite?
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