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 $[.]$ 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 $8$

Given that the range of $f\left(x\right)$ is $[2,6,5]$

So the greatest value of $f\left(x\right)=6$

The maximum value of $\cos x$ is $1$

Now given that $h\left(x\right)=\left[\frac{\cos x+f\left(x\right)}{\lambda }\right]$ is continuous.

Since $h\left(x\right)$ is step function, it should be equal to constant for it to be continuous.

So $h\left(x\right)=0$ , the maximum value of $\cos x+f\left(x\right)$ is $1+6=7$

Therefore the minimum value of $\lambda$ should be greater than $7$ for $h\left(x\right)=0$

So the least value of $\lambda$ is $8$