Polynomials

Question

Polynomials

anonymous · Answer

f(x) is a Polynomial with real coefficients
find the equation of f(x) if 

$$f(2x)=f'(x) f''(x)$$

anonymous · Answer

f(x) should be an exponential function

anonymous · Answer

Let's suppose that degree of f(x) is n
HENCE:
degree of f(2x) is : n
degree of f'(x) is : n-1
degree of f''(x) is : n-2
degree of f'(x)f''(x) is : n-1+n-2=2n-3

According to the equation degree of f(2x) must equal to degree of f'(x)f''(x)
then
n=2n-3 -->  n=3
so we have
$$f(x)=ax^3+bx^2+cx+d \ f(2x)=8ax^3+4bx^2+2cx+d \ f'(x)=3ax^2+2bx+c\f''(x)=6ax+2b$$

According to the equation
 $$f(2x)=f'(x) f''(x)$$
then u must have
$$f(2x)=8ax^3+4bx^2+2cx+d=(3ax^2+2bx+c)(6ax+2b)=f'(x) f''(x)$$
that gives
a=4/9 and b=c=d=0 
then the only answer is 
$$f(x)=\frac{4}{9}x^3$$