Question

The values of $\lambda$ so that $f(x) = \sin x - \cos x - \lambda x + C$ decreases for all real values of $x$ are :

• A. $1<\lambda<\sqrt{2}$
• B. $\lambda \geq 1$
• C. $\lambda \geq \sqrt{2}$
• D. $\lambda<1$

Hint:

For functions with trigonometric terms, use the maximum value of the trigonometric expression to determine the range of constants for monotonicity.

Answer 1

The Correct Option is A

To ensure that the function \( f(x) \) is decreasing for all real values of \( x \), its derivative must satisfy \( f'(x) \leq 0 \) for all \( x \).

The derivative of the function is:
\[ f'(x) = \cos x + \sin x - \lambda \]
For \( f'(x) \leq 0 \) to hold for all \( x \), the maximum value of \( \cos x + \sin x \) must be less than or equal to \( \lambda \).

The maximum value of \( \cos x + \sin x \) is \( \sqrt{2} \). Therefore,
\[ \cos x + \sin x \leq \sqrt{2} \]

Hence, for the derivative to be non-positive for all \( x \),
\[ \lambda \geq \sqrt{2} \]

Final Answer:
The function \( f(x) \) is decreasing for all real \( x \) when \[ \boxed{\lambda \geq \sqrt{2}} \]