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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]} $. 📰 This Office Shattered Outlook Notes Will Shock Youspot The Hidden Crises Before Its Too Late 3682151 📰 Torrentz Download 9857659 📰 Unexpected News How To Transfer Pictures From Android To Computer And The Truth Shocks 📰 Tori Spellings Secret Million Dollar Net Worth Revealed 7296155 📰 Sources Say Data Type Clob In Oracle And The World Is Watching 📰 Parasite Ending 📰 If A Person Invests 2000 At An Annual Interest Rate Of 5 Compounded Annually How Much Will The Investment Be Worth After 3 Years 8008724 📰 Crack Laxcrossword Like A Prohidden Secrets Revealed Instantly 9735935 📰 Worst Dad Jokes Of All Time 📰 Discover The Best Free Online Golf Games You Can Play Instantlyno Downloads Required 892809 📰 50265482 7772237 📰 Nicks 5822233 📰 Bloons Tower 📰 A Science Communicator Creates A Video On Carbon 14 Dating Explaining That The Isotope Decays With A Half Life Of 5730 Years If A Fossil Contains 25 Of Its Original Carbon 14 How Many Half Lives Have Passed And What Is The Approximate Age Of The Fossil 8419267 📰 Stock To Buy Today