y = sin x + cos x ⇔ √ 2 ( sin x . √ 2 2 + cos x . √ 2 2 ) ⇔ √ 2 ( sin x . cos π 4 + cos x . sin π 4 ) ⇔ √ 2 sin ( x + π 4 ) Ta có: − 1 ≤ sin ( x + π 4 ) ≤ 1 ⇔ − √ 2 ≤ √ 2 sin ( x + π 4 ) ≤ √ 2 M a x y = √ 2 khi sin ( x + π 4 ) = 1 ⇔ ( x + π 4 ) = π 2 + k 2 π M i n y = − √ 2 khi sin ( x + π 4 ) = − 1 ⇔ ( x + π 4 ) = − π 2 + k 2 π ( k ∈ Z ) .
`y = sinx + cosx`
`<=> \sqrt2(\sin x.\sqrt2/2 + \cos x .\sqrt2/2)`
`<=> \sqrt2(\sin x.\cos \frac{\pi}{4} + \cos x. sin \frac{\pi}{4} )`
`<=> \sqrt2sin(x + \pi/4) `
Ta có:
`-1 ≤ \sin (x + \pi/4) ≤1`
`⇔ -\sqrt2 ≤ \sqrt2sin(x + \pi/4) ≤\sqrt2`
`\Max_{y}=\sqrt{2}` khi `\sin (x+\frac{\pi}{4})=1 \Leftrightarrow (x+\frac{\pi}{4})=\frac{\pi}{2}+k2\pi`
$Min_{y}=-\sqrt{2}$ khi `\sin (x+\frac{\pi}{4})=-1 \Leftrightarrow (x+\frac{\pi}{4})=-\frac{\pi}{2}+k2\pi`
`(k\in\mathbb Z)`.
y = sin x + cos x ⇔ √ 2 ( sin x . √ 2 2 + cos x . √ 2 2 ) ⇔ √ 2 ( sin x . cos π 4 + cos x . sin π 4 ) ⇔ √ 2 sin ( x + π 4 ) Ta có: − 1 ≤ sin ( x + π 4 ) ≤ 1 ⇔ − √ 2 ≤ √ 2 sin ( x + π 4 ) ≤ √ 2 M a x y = √ 2 khi sin ( x + π 4 ) = 1 ⇔ ( x + π 4 ) = π 2 + k 2 π M i n y = − √ 2 khi sin ( x + π 4 ) = − 1 ⇔ ( x + π 4 ) = − π 2 + k 2 π ( k ∈ Z ) .