Question

If the function $f\left(x\right)=\left[\frac{(x-5{)}^3}{A}\right]\sin (x-5)+a\cos (x-2)$ where $[.]$ denotes the greatest integer function,is continuous and differentiable in (7,9),then:

• A. $A\in [8,64]$
• B. $A\in (0,8]$
• C. $A\in (64,\infty )$
• D. $A\in [8,16]$

Answer 1

The correct option is C $A\in (64,\infty )$

$f\left(x\right)=\left[\frac{(x-5{)}^3}{A}\right]\sin (x-5)+a\cos (x-2)$
$\left[x\right]$ is not continuous and differentiable at integral values.
So f(x) will be continuous and differentiable in [7, 9] if $\left[\frac{(x-5{)}^3}{A}\right]=0$
$\Rightarrow 0\le \frac{(x-5{)}^3}{A}<1$
$0\le \frac{(x-5{)}^3}{A}$ is true for $A>0$
Now, if $\frac{(x-5{)}^3}{A}<1$, then $A>{(9-5)}^3$ (since we are checking in the interval $(7,9)$)
$\Rightarrow A>64$
$\therefore Aϵ(64,\infty )$.