Question

Let $f\left(x\right)=(x^3+2{)}^{30}$. If $f^n\left(x\right)$ is a polynomial of degree $20$, where $f^n\left(x\right)$ denotes the $nth$ derivative of $f\left(x\right)$ w.r.t.$x$, then the value of $n$ is-

• A. $60$
• B. $40$
• C. $70$
• D. $None$

Answer 1

The correct option is A $60$

$f\left(x\right)=(x^3+2{)}^{30}$

$f^n\left(x\right)=$ polynomial of degree $20$

$f^n\left(x\right)=n^{th}$ derivative of $f\left(x\right)$

$f\left(x\right){=}^{30}{C}_0\left(x^3{)}^0\right({)}^{30}{+}^{30}{C}_0\left(x^4\right)(2{)}^{30}$

${}^{30}{C}_0\left(x^3{)}^r\right(2{)}^{30-r}$

$x^{3r}=r^{21}$

$r=7$

${=}^{30}{C}_7(x^3{)}^7{2}^{21}$

$=30\times 2$

$=60$