Question

If $f\left(x\right)=\left|\begin{array}{ccc}{x}^n& sinx& cosx\\ n!& sin\left(\frac{n\pi }{2}\right)& cos\left(\frac{n\pi }{2}\right)\\ a& a^2& a^3\end{array}\right|$

then the value of $\frac{d^{n}f}{d{x}^n}$ at $x=0$, is equal to

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

Answer: (B)

$0$

$f\left(x\right)=x^n(a^{3}sin\frac{n\pi }{2}-a^{2}cos\frac{n\pi }{2})-si{n}^x(n!a^3-acos\frac{n\pi }{2})+cosx(n!a^2-asin\frac{n\pi }{2})$

$f^{|}\left(x\right)=n{x}^{n-2}(a^{3}sin\frac{n\pi }{2}-a^{2}cos\frac{n\pi }{2})-cosx(n!a^3-acos\frac{n\pi }{2})-sinx(n!a^2-asin\frac{n\pi }{2})$

$f^{||}\left(x\right)=n^{(n-1)}{x}^{n-2}(a^{3}sin\frac{n\pi }{2}-a^{2}cos\frac{n\pi }{2})+sinx(n!a^3-acos\frac{n\pi }{2})-cosx(n!a^2-asin\frac{n\pi }{2})$

$f^n\left(0\right)=n!(a^{3}sin\frac{n\pi }{2}-a^{2}cos\frac{n\pi }{2})-\left(sin\frac{n\pi }{2}\right)(n!a^3-acos\frac{n\pi }{2})+\left(cos\frac{n\pi }{2}\right)(n!a^{2}asin\frac{n\pi }{2})$
$=0$