If sin^4 α+cos^4 β = 4sin α cos β , 0≤α,β ≤ (π/2) , then sin α+cos β equal to …


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Question Number 138099 by liberty last updated on 10/Apr/21

$I f \sin^{4} α + \cos^{4} β = 4 \sin α \cos β,$ $0 ⩽ α, β ⩽ \frac{π}{2}, t h e n \sin α + \cos β$ $e q u a l t o \dots$
Commented by mr W last updated on 10/Apr/21

$r e a s o n :$ $s a y p = \sin α, q = \cos β$ $f o r a g i v e n q, y o u c a n s o l v e f o r p :$ $p^{4} - 4 q p + q^{4} = 0$ $t h a t m e a n s p + q \neq c o n s t a n t$
Commented by mr W last updated on 10/Apr/21

$n o u n i q u e v a l u e!$ $0 ⩽ \sin α + \cos β ⩽ 1.25099$ $s o t h e q u e s t i o n s h o u l d h a v e b e e n :$ $f i n d t h e m a x i m u m v a l u e o f$ $\sin α + \cos β .$
Commented by liberty last updated on 10/Apr/21

$a n s w e r i s 2$
Commented by mr W last updated on 10/Apr/21

$w r o n g!$ $i f \sin α + \cos β = 2, t h e n$ $\sin α = 1, \cos β = 1 a n d$ $\sin^{4} α + \cos^{4} β = 2 \neq 4 \sin α \cos β = 4$
Commented by mr W last updated on 10/Apr/21

$y o u c a n a l s o s e e :$ $w i t h α = 0, β = \frac{π}{2} w h i c h f u l f u l l$ $\sin^{4} α + \cos^{4} β = 4 \sin α \cos β, b u t$ $\sin α + \cos β = 0 \neq 2 .$ $a n o t h e r e x a m p l e :$ $α = π / 3, β = 1.40749 w h i c h f u l f u l l$ $\sin^{4} α + \cos^{4} β = 4 \sin α \cos β, b u t$ $\sin α + \cos β = 1.0286 \neq 2 .$ $t h e s e e x a m p l e s s h o w a l s o t h a t$ $\sin α + \cos β m a y h a v e d i f f e r e n t v a l u e s .$
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