Question

If $y=[x{]}^{4+\left\{x\right\}}+\{x{\}}^{\left[x\right]-3},$ then $\frac{dy}{dx}$ at $x=\pi$ is
$($Where $[\cdot ]$ denotes the greatest integer function and $\{\cdot \}$ denotes the fractional part function$)$

• A. $3\cdot 3^3\ln 3$
• B. $3\cdot 3^{\pi }\ln \pi$
• C. $3\cdot 3^3\ln \pi$
• D. $3\cdot 3^{\pi }\ln 3$

Answer 1

The correct option is D $3\cdot 3^{\pi }\ln 3$

$y=[x{]}^{4+\left\{x\right\}}+\{x{\}}^{\left[x\right]-3}=[x{]}^{4+(x-[x\left]\right)}+(x-\left[x\right]{)}^{\left[x\right]-3}$
$\Rightarrow y=(3{)}^{4+(x-3)}+(x-3{)}^0$
$(\because [3.14]=3,\{x\}=x-[x]=x-[3.14]=x-3)$
$\Rightarrow y=3^{1+x}+1=3\cdot 3^x+1\Rightarrow {\frac{dy}{dx}|}_{x=\pi }=3\cdot 3^{\pi }\ln 3$