Chứng minh rằng forall n in mathbb N ^ * forall (x, y) in( mathbb R + )^ 2 , root(x + y, n) <= root(x, n) + root(y, n)

To prove the inequality:

\[
\forall n \in \mathbb{N}^*, \forall (x, y) \in (\mathbb{R}^+)^2, \sqrt[n]{x + y} \leq \sqrt[n]{x} + \sqrt[n]{y},
\]

we can proceed using the following steps.

### 1. Define the Functions
Let \( f_n(x) = \sqrt[n]{x} \), where \( n \in \mathbb{N}^* \) and \( x \in \mathbb{R}^+ \). The inequality we want to prove is:

\[
f_n(x + y) \leq f_n(x) + f_n(y).
\]

### 2. Apply Minkowski's Inequality
Minkowski's inequality in \( L^p \) space (for \( p = \frac{1}{n} \), where \( n > 0 \)) states that for any non-negative \( a_1, a_2, \dots, a_m \) and \( b_1, b_2, \dots, b_m \), the following inequality holds:

\[
\left(\sum_{i=1}^m (a_i + b_i)^p \right)^{\frac{1}{p}} \leq \left(\sum_{i=1}^m a_i^p \right)^{\frac{1}{p}} + \left(\sum_{i=1}^m b_i^p \right)^{\frac{1}{p}}.
\]

For \( m = 2 \) and \( p = \frac{1}{n} \), consider \( a_1 = x^{\frac{1}{n}} \) and \( b_1 = y^{\frac{1}{n}} \). Then:

\[
(x^{\frac{1}{n}} + y^{\frac{1}{n}})^n \geq x + y.
\]

Taking the \( n \)-th root on both sides:

\[
\sqrt[n]{x + y} \leq \sqrt[n]{x} + \sqrt[n]{y}.
\]

### 3. Consider the Equality Case
Equality holds if and only if \( x = 0 \), \( y = 0 \), or \( x^{\frac{1}{n}} \) and \( y^{\frac{1}{n}} \) are proportional.

### Conclusion
Thus, the inequality is proven:

\[
\forall n \in \mathbb{N}^*, \forall (x, y) \in (\mathbb{R}^+)^2, \sqrt[n]{x + y} \leq \sqrt[n]{x} + \sqrt[n]{y}.
\]

This concludes the proof.

Would you like further details or have any questions?

Here are some related questions to explore:

1. How can you apply Minkowski's inequality to higher dimensions?
2. What are other important inequalities in \( L^p \) spaces?
3. How does Minkowski's inequality relate to the triangle inequality?
4. What is the significance of the equality condition in Minkowski's inequality?
5. How does this proof change if \( x \) and \( y \) can be negative?

**Tip:** Minkowski's inequality generalizes the triangle inequality for vector spaces, offering insights into the geometry of \( L^p \) spaces.

Learn how to prove the inequality \(\forall n \in \mathbb{N}^*, \forall (x, y) \in (\mathbb{R}^+)^2, \sqrt[n]{x + y} \leq \sqrt[n]{x} + \sqrt[n]{y}\) with a detailed step-by-step solution using Minkowski's inequality. This problem covers key mathematical concepts including inequalities, roots, and applications of Minkowski's inequality.

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