ab = \frac{1}{2} \quad \Rightarrow \quad \frac{1}{a^2} + \frac{1}{b^2} = \frac{(a^2 + b^2)^2}{a^2 b^2} = \frac{1}{(\frac{1}{2})^2} = 4
Related Articles
- We apply the AM-GM inequality to \(\frac{1}{a^2} + \frac{1}{b^2}\):
- \frac{1}{a^2} + \frac{1}{b^2} \geq \frac{2}{ab}
- Given \(a^2 + b^2 = 1\), the maximum of \(ab\) is \(\frac{1}{2}\) when \(a = b = \frac{\sqrt{2}}{2}\). Then:
- Therefore, the expression becomes:
- + 4 = 9
- Verification at \(x = \frac{\pi}{4}\) shows \(\cos x = \sin x = \frac{\sqrt{2}}{2}\), yielding:
Trending Articles
Roblox Piano Tiles Glitch Lets You Skip Every Song
This Warframe Mag Skin Is Hiding A Gamebreaking Trick
Why Are Warframe Mag Skins Suddenly So Legendary?
Unlock The Secret Top 5 Mag Skins In Warframe
The Rarest Warframe Mag Skin You Can Actually Get
Can You Survive the Brutal Stellaris Liberation Wars?
Unlock the Secret Achievement 3 Drill Test Cap—Can You Beat It?
Achievement 3 Drill Test Cap Glitch Found: See What Happens!
This Chicken Casserole is OP: How Do You One-Up Your Broccoli?
The Abyssus Disc Thrower is Broken: Watch This Gameplay Instantly