Question

The value of $k$ in order that $f\left(x\right)=\sin x-\cos x-kx+5$ decreases for all positive real values of $x$ is given by

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

Answer 1

The correct option is C $k>\sqrt{2}$

For $f\left(x\right)$ to decrease, $f^{\prime }\left(x\right)<0$
$\Rightarrow f^{\prime }\left(x\right)=\cos x+\sin x-k<0\Rightarrow k>\sin x+\cos x$
$\Rightarrow k>\sqrt{2}$ as the maximum value of $\sin x+\cos x$ is $\sqrt{2}$