Question

The value of $b$ for which function $f\left(x\right)=\sin x-bx+c$ is decreasing in the interval $\left(-\infty ,\infty \right)$ is given by

• A. $b<1$
• B. $b\ge 1$
• C. $b>1$
• D. $b\le 1$

Answer 1

The correct option is B

$b\ge 1$

Explanation for the correct option:

Find the value of $b$:

A function $f\left(x\right)=\sin x-bx+c$ is given.

It is also given that the given function is a strictly decreasing function.

We know that, if a function is a decreasing function, then the derivative of the function is less than zero.

Therefore, $f\text{'}\left(x\right)\le 0$.

$$\begin{gathered} \cos x-b\le 0 \\ \Rightarrow \cos x\le b \end{gathered}$$

Since, $-1\le \cos x\le 1$.

So, for strictly decreasing function $b\ge 1$.

Therefore, The value of $b$ in order that $f\left(x\right)=\sin x-bx+c$ decreases in the interval $\left(-\infty ,\infty \right)$ is given by $b\ge 1$.

Hence, option (B) is the correct answer.