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If \(f:\mathbb{R}\to\mathbb{R}\) is given by \(f(x)=\sin x+\cos x\), what is the range of (f)?यदि \(f:\mathbb{R}\to\mathbb{R}\) को \(f(x)=\sin x+\cos x\) से दिया गया है, तो (f) का परास क्या है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: \([-\sqrt{2},\sqrt{2}]\)\([-\sqrt{2},\sqrt{2}]\)
Step 1: Write \(\sin x+\cos x=\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)\). Step 2: The sine function lies in ([-1,1]). Step 3: Hence the range is \([-\sqrt{2},\sqrt{2}]\).चरण 1: \(\sin x+\cos x=\sqrt{2}\sin\left(x+\frac{\pi}{4}\right)\) लिखा जा सकता है। चरण 2: (\sin) का मान ([-1,1]) में रहता है। चरण 3: इसलिए परास \([-\sqrt{2},\sqrt{2}]\) है।
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