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. $A\in (0,0)$

Answer 1

Answer: (C)

$A\in [64,\infty )$

$\left[x\right]$ is not continuous and differentiable at integral values (points).

So $f\left(x\right)$ is continuous and differentiable in $(7,9)$.

if $\left[\frac{\left(\right(x-5{)}^3)}{A}\right]=0$

$\Rightarrow A\ge (9-5{)}^3\ge 64$

$A\in [64,\infty )$ ,$\forall x\in (7,9)$