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Prove that if f is any cubic polynomial function f(x)=ax3+bx2+cx+dthen the coefficients of f are completely determined by the values of f(x) and its derivative at x=0 as follows

a=f"'(0)6;b=f"(0)2;c=f'(0);d=f(0)

Short Answer

Expert verified

We determine the values of a, b, c, d in terms of f and its derivatives at x=0

Step by step solution

01

Given information

We are given a cubic polynomialf(x)=ax3+bx2+cx+d

02

Find the derivatives of the function

We have,

f(x)=ax3+bx2+cx+df'(x)=3ax2+2bx+cf"(x)=6ax+2bf"'(x)=6a

Now compute all the values at x=0

We get,

f(0)=d(1)f'(0)=c(2)f"(0)=2bb=f"(0)2(3)f"'(0)=6aa=f"'(0)6(4)

We determine the values of a, b, c, d in terms of f and its derivatives at x=0

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