Question

Let $f\left(x\right)$ be a continuous function whose range is $[2,6.5]$. If $h\left(x\right)=\left[\frac{\cos x+f\left(x\right)}{\lambda }\right],\lambda \in N$, be continuous, where $[\cdot ]$ denotes the greatest integer function, then the least value of $\lambda$ is

• A. $6$
• B. $7$
• C. $8$
• D. None of these

Answer 1

The correct option is C $8$

$h\left(x\right)$ is continuous. Hence, $h\left(x\right)$ needs to be a constant function attaining an integral value.
The term inside the greatest integer function must have values between two consecutive integers.
We have $1\le \cos x+f\left(x\right)\le 7.5$ . Since, $\lambda$ is an integer and the range of $\frac{(\cos x+f(x\left)\right)}{\lambda }$ must be between two successive integers, it must be between $0$ and $1$. Hence, the least value of $\lambda$ is $8$.