How to Prove the AM-GM Inequality for Numbers with Product of 1

Hi there! This is a beautiful inequality problem that combines algebraic manipulation with some clever insights. Let's work through it together! 🌟

1. What We're Solving:

We need to prove that for non-negative real numbers a and b where ab = 1, the expression (a+b)/(1+a²) + (a+b)/(1+b²) is always at least 2.

2. The Approach:

Here's our game plan! Since we have the constraint ab = 1, we can use substitution to simplify our work. We'll substitute b = 1/a (since ab = 1 means b = 1/a when a ≠ 0), then work with a single variable. This transforms a two-variable problem into a more manageable one-variable optimization problem.

3. Step-by-Step Solution:

Step 1: Use the constraint ab = 1 Since ab = 1 and a,b ≥ 0, we have b = 1/a (note: a > 0 since if a = 0, then ab = 0 ≠ 1).

Step 2: Substitute b = 1/a into our inequality Our expression becomes: (a + 1/a)/(1 + a²) + (a + 1/a)/(1 + (1/a)²)

Step 3: Simplify the second fraction Notice that 1 + (1/a)² = 1 + 1/a² = (a² + 1)/a²

So our expression is: (a + 1/a)/(1 + a²) + (a + 1/a)/((a² + 1)/a²)

The second term becomes: (a + 1/a) × a²/(a² + 1) = a²(a + 1/a)/(a² + 1)

Step 4: Factor out common terms Our expression is now: (a + 1/a)/(1 + a²) + a²(a + 1/a)/(1 + a²)

Factor out (a + 1/a)/(1 + a²): = (a + 1/a)/(1 + a²) × (1 + a²) = (a + 1/a)

Wait, let me recalculate this more carefully...

Actually, let's be more systematic: (a + 1/a)/(1 + a²) + (a + 1/a)/(1 + 1/a²)

The second denominator: 1 + 1/a² = (a² + 1)/a²

So we have: (a + 1/a)/(1 + a²) + (a + 1/a) × a²/(a² + 1)

= (a + 1/a)/(a² + 1) + a²(a + 1/a)/(a² + 1)

= (a + 1/a)(1 + a²)/(a² + 1) = a + 1/a

Step 5: Apply the AM-GM inequality Now we need to show that a + 1/a ≥ 2 for a > 0.

By the AM-GM inequality: (a + 1/a)/2 ≥ √(a × 1/a) = √1 = 1

Therefore: a + 1/a ≥ 2

Step 6: Equality condition Equality occurs when a = 1/a, which means a² = 1, so a = 1 (since a > 0). When a = 1, we have b = 1, and both sides of our original inequality equal 2.

4. The Answer:

The inequality (a+b)/(1+a²) + (a+b)/(1+b²) ≥ 2 is proven! ✅

The key insight was recognizing that the constraint ab = 1 allows us to reduce this to showing a + 1/a ≥ 2, which follows directly from AM-GM.

5. Memory Tip:

Remember this pattern: When you see ab = 1 in a constraint, try substituting b = 1/a to reduce variables. Also, whenever you see expressions like a + 1/a, think AM-GM inequality - it's often the key to proving the bound! The constraint ab = 1 creates a beautiful symmetry that simplifies complex-looking expressions.

Great job working through this challenging problem! The combination of substitution and AM-GM is a powerful technique you'll see again in many inequality proofs. 🎯