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📰 Thus, the minimum value is $ \boxed{0} $. 📰 Question: Find the range of the function $ f(x) = \frac{1}{\sin^2 x + 4\cos^2 x} $ as $ x $ ranges over all real numbers such that $ \sin^2 x + \cos^2 x = 1 $. 📰 Solution: Since $ \sin^2 x + \cos^2 x = 1 $, we write $ f(x) = \frac{1}{\sin^2 x + 4\cos^2 x} = \frac{1}{1 + 3\cos^2 x} $. Let $ y = \cos^2 x $, so $ 0 \leq y \leq 1 $. Then $ f(x) = \frac{1}{1 + 3y} $. As $ y $ varies from 0 to 1, $ 1 + 3y $ varies from 1 to 4, so $ f(x) $ varies from $ \frac{1}{4} $ to $ 1 $. Hence, the range of $ f(x) $ is $ \left[\frac{1}{4}, 1\right] $. The final answer is $ \boxed{\left[\frac{1}{4}, 1\right]} $. 📰 Liven Reviews 8581546 📰 Love The Long Hoco Dress With Sleeves These Stellar Styles Are Copycat Versions You Cant Miss 7589981 📰 He Was Called Fuck Ass Bob Watch The Reaction That Shook The Internet 4065900 📰 Financing For Vehicles 9853484 📰 Railworks Game 📰 Unlock Endless Funplay Free Arcade Games Online Today With No Cost 8982090 📰 Zeta Charter School 2888843 📰 This Simple Recipe Was Forbidden In Casareccenow Watch The Chaos Unfold 7327618 📰 Study Reveals Free Gaming And The Response Is Massive 📰 Sudden Change How Can You Start A Business Without Money And Officials Respond 📰 Asura Wrath The Dark Force That Will Destroy All Hope 9010617 📰 Fiorentinas Surprise Lineup Vs Man United The Battle That Defies Every Prediction 4337727 📰 This Rare Bat Pokmon Will Blow Your Pokdex Awayheres Why 2907976 📰 Apply Online Bank Account 📰 2 Open Roth Ira Fidelity The Secret Tax Move That Saves Thousands Instantly 1976310