F(x) is a function which is defined as F(x) = (x+1)² * e^2x. The fiftieth derivative of F(x) with respect to x at x=0, i.e. F⁽⁵⁰⁾(0), is expressible in the form k*2ⁿ where k is an odd integer and n is a positive integer.

Determine the respective values of k and n.

now the mth derivative of f(x) takes the form (a*x^2+b*x+c)*e^(2x) so if we let f(m,x) be the mth then
f(m,x)=( a(m)*x^2+b(m)*x+c(m) ) * e^(2x) so then
f(50,0) = c(50)

now if f(m+1,x) = d/dx f(m,x) and thus
f(m+1,x) = ( 2*a(m)*x^2 + 2*(a(m)+b(m))*x + b(m)+2c(m) )*e^(2x) thus
a(m+1) = 2*a(m) and a(0)=1 thus a(m) = 2^m

b(m+1) = 2*a(m)+2b(m) and b(0)=2 thus b(m) = 2^(m+2)

c(m+1) = b(m) + 2c(m)  and c(0) = 1 thus
c(m) = (4m + 1) * 2^m

thus f(50,0) = (4*50 + 1) * 2^50
f(50,0) = 201*2^50
thus k=201 and n=50

Posted by Daniel on 2009-12-18 13:35:14