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Chapter 2: Q. 3 (page 200)

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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Most popular questions from this chapter

In Exercises 69鈥�80, determine whether or not f is continuous and/or differentiable at the given value of x. If not, determine any left or right continuity or differentiability. For the last four functions, use graphs instead of the definition of the derivative.

fx=x2-3,ifx<3x+2,ifx鈮�3,x=3

Use (a) the h鈫�0definition of the derivative and then

(b) the z鈫�cdefinition of the derivative to find f'(c) for each function f and value x=c in Exercises 23鈥�38.

28.f(x)=x4+1,x=2

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=3x+1x2+1

Find the derivatives of the functions in Exercises 21鈥�46. Keep in mind that it may be convenient to do some preliminary algebra before differentiating.

f(x)=x+45

Prove that if f is a quadratic polynomial function f(x)=ax2+bx+cthen the coefficient of f are completely determined by the values of f(x) and its derivatives at x=0 as follows

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