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 $f(x)$ decreases for all real $x$, the derivative of $f(x)$ must be negative for all $x$. \[ f'(x) = \cos x + \sin x - \lambda \] For $f'(x) \leq 0$ for all $x$, we need the maximum value of $\cos x + \sin x$ to be less than or equal to $\lambda$. The maximum value of $\cos x + \sin x$ is $\sqrt{2}$, so: \[ \lambda \geq \sqrt{2} \] Thus, the correct range of $\lambda$ is $1<\lambda<\sqrt{2}$.