(X^2)+(9Y^2)=10,MAX((X+3Y)^2)= by AM–GM inequality

To solve this problem using the Arithmetic Mean–Geometric Mean (AM–GM) inequality, we aim to find the maximum value of \((X + 3Y)^2\) under the given constraint \((X^2) + (9Y^2) = 10\).

### Step 1: Restate the Problem

We are given the equation:

\[
X^2 + 9Y^2 = 10
\]

We need to maximize the expression:

\[
(X + 3Y)^2
\]

### Step 2: Apply AM–GM Inequality

The AM–GM inequality states that for any non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean.

To apply this concept, notice that:

\[
X^2 + 9Y^2 = 10
\]

We can rewrite \( X^2 + 9Y^2 \) in terms of squares of sums, noting that this is the sum of two squares (similar to how AM-GM works):

The AM-GM inequality doesn't directly apply here as a shortcut for maximizing \( (X + 3Y)^2 \), but we can solve it using standard maximization techniques like Lagrange multipliers or substitution methods for now.

### Step 3: Parametrize the Problem Using Trigonometry (Alternate Approach)

One effective approach is to parametrize \(X^2 + 9Y^2 = 10\) by substituting trigonometric identities. We can let:

\[
X = \sqrt{10} \cos \theta, \quad Y = \frac{\sqrt{10}}{3} \sin \theta
\]

Thus, we want to maximize:

\[
(X + 3Y)^2 = (\sqrt{10} \cos \theta + \sqrt{10} \sin \theta)^2 = 10 (\cos \theta + \sin \theta)^2
\]

Using the identity \(\cos \theta + \sin \theta = \sqrt{2} \cos \left( \theta - \frac{\pi}{4} \right)\), we get:

\[
(X + 3Y)^2 = 10 \times 2 = 20
\]

Thus, the maximum value of \((X + 3Y)^2\) is \(20\).

### Conclusion:

The maximum value of \((X + 3Y)^2\) is \(\boxed{20}\).

Would you like further clarification on any steps? Here are some related questions you might find helpful:

1. How do trigonometric parametrizations work in maximizing constrained expressions?
2. What is the AM–GM inequality, and how can it be applied in different cases?
3. How would you solve a constrained optimization problem using Lagrange multipliers?
4. What other inequalities are useful in mathematical optimization problems?
5. How does the identity \(\cos \theta + \sin \theta = \sqrt{2} \cos \left( \theta - \frac{\pi}{4} \right)\) help in solving such problems?

**Tip**: When maximizing a quadratic expression under a constraint, consider using trigonometric substitution for a cleaner solution.

(X^2) + (9Y^2) = 10, MAX((X + 3Y)^2) using AM–GM inequality

Solve the problem of maximizing (X + 3Y)^2 under the constraint (X^2) + (9Y^2) = 10 using the AM–GM inequality. This solution includes advanced algebra, maximization techniques, and trigonometric parametrization. Ideal for undergraduate students or advanced high school learners.

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