Question

if $f\left(x\right)=\sin x-\cos x-ax+b$ decreases for all $x\in R$ then

• A. $a<1$
• B. $a>1$
• C. $a<\sqrt{2}$
• D. $a>\sqrt{2}$

Answer 1

The correct option is D $a>\sqrt{2}$

$f,\left(x\right)=\cos x+\sin x-a$
$f,\left(x\right)=\sqrt{2}\left(\sin \right(\frac{\pi }{4}+x\left)\right)-a$
$-\sqrt{2}<\sqrt{2}\sin (\frac{\pi }{4}+x)<\sqrt{2}$ for $\forall x\in R$
$-\sqrt{2}-a<f^1\left(x\right)<\sqrt{-}a$
for $f,\left(x\right)<0$ $a>\sqrt{2}$