Question

The value of \(k\) such that the function \(f(x) = sinx-cosx-kx+b\) is strictly decreasing for all real \(x\) is

• A. \(k < 1\)
• B. \(k > 1\)
• C. \(k < \sqrt{2}\)
• D. \(k > \sqrt{2}\)

Hint:

f decreasing for all x needs f prime negative for all x; the maximum of cos x + sin x is sqrt2.

Answer 1

The Correct Option is D

Step 1: Bound the derivative:
By Cauchy-Schwarz, $|\cos x+\sin x|\leq\sqrt{1^2+1^2}=\sqrt2$, with equality when $\tan x=1$.

Step 2: Apply the requirement:
For $f'<0$ at the worst case we need $\sqrt2-k<0$, so $k>\sqrt2$.

Step 3: Pick:
Option D.

Final Answer:
The maximum of cos x + sin x is sqrt 2, so k must exceed it. \[ \boxed{\text{(D) }k>\sqrt2} \]