Question

Let f(x)| be a quadratic function such that f(0) = -4| and integral f(x) / x^2(x + 3)^6 dx| is a rational function. Determine the value of f'(0)|.

Let f(x) be a quadratic function such that f(0)=?4 and ?f(x)/x2(x+3)^6dx is a rational function.

Answer 1

i)

The function f(x) is a quadratic function and is of the form

$f(x)=ax^2+bx+c$

Given

$f(0)=-4 => a(0)^2+b(0)+c=-4 => c=-4$

This implies,

$f(x)=ax^2+bx-4$

ii)

The integral is

$\int \frac{f(x)}{x^2(x+3)^6}dx$

Consider the partial factor decomposition of the function in the integration

$\\ \frac{ax^2+bx-4}{x^2(x+3)^6}=\frac{A}{x}+\frac{B}{x^2}+\frac{C}{(x+3)}+\frac{D}{(x+3)^2}+\frac{E}{(x+3)^3}+\frac{F}{(x+3)^4}+\frac{G}{(x+3)^5}+\frac{H}{(x+3)^6}$

Integrating after partial factor decomposition would make the resulting integral a rational function only if A=0, C=0 as the integral of 1/x and 1/(x+3) is ln(x) and ln(x+3) respectively

Therefore,

$\\ \frac{ax^2+bx-4}{x^2(x+3)^6}=\frac{B}{x^2}+\frac{D}{(x+3)^2}+\frac{E}{(x+3)^3}+\frac{F}{(x+3)^4}+\frac{G}{(x+3)^5}+\frac{H}{(x+3)^6}$

Multiply both sides by x^2(x+3)^6

$\\ ax^2+bx-4=B(x+3)^6+D(x^2)(x+3)^4+E(x^2)(x+3)^3+F(x^2)(x+3)^2+G(x^2)(x+3)+H(x^2)$

For the above expression on the RHS, evaluate using Pascals Triangle

power=0 1
power=1 1 1
power=2 1 2 1
power=3 1 3 3 1
power=4 1 4 6 4 1
power=5 1 5 10 10 5 1
power=6 1 6 15 20 15 6 1

$\\ (x+3)^6=x^6(3)^0+6x^5(3^1)+15 x^4(3^2)+20 x^3(3^3)+15x^2(3^4)+6 x(3^5)+1(3^6) \\ \\ (x+3)^6=x^6+18 x^5+135 x^4+540 x^3+1215 x^2+1458 x+729$

$\\ (x+3)^4=x^4+4 x^3(3^1)+6 x^2(3^2)+4 x^1(3^3)+x^0(3^4)=x^4+12x^3+54x^2+108x+81$

$\\ (x+3)^3=x^3+3 x^2(3)^1+3 x^1(3)^2+x^0(3^3)=x^3+9x^2+27x+27$

$\\ (x+3)^2=x^2+6x+9$

The expression on the RHS becomes

$\\ ax^2+bx-4=B[x^6+18 x^5+135 x^4+540 x^3+1215 x^2+1458 x+729]+D(x^2)[x^4+12x^3+54x^2+108x+81]+E(x^2)[x^3+9x^2+27x+27]+F(x^2)[x^2+6x+9]+G(x^2)(x+3)+H(x^2)$

That is

$\\ ax^2+bx-4=B[x^6+18 x^5+135 x^4+540 x^3+1215 x^2+1458 x+729]+D(x^2)[x^4+12x^3+54x^2+108x+81]+E(x^2)[x^3+9x^2+27x+27]+F(x^2)[x^2+6x+9]+G(x^2)(x+3)+H(x^2)$

The terms of x only exist for the first expression as all other expression are multiplied by x^2. This implies, coefficient of x is

$1458B$

The constant term only exist for the first expression as all other expression are multiplied by x^2. This implies, constant term is

$729B$

The terms of x^2 exist for the first expression and all other expressions multiplied by constant term. This implies, coefficient of x^2 is

$[1215B+81D+27E+9F+3G+H]$

Equate the coefficients of x, x^2 and constant term implies

$729B=-4 => B=\frac{-4}{729}$

$b=1458B=1458 \times \frac{-4}{729}=-8$

$\\ a=[1215(\frac{-4}{729})+81D+27E+9F+3G+H]=[\frac{-5}{3}+81D+27E+9F+3G+H]$

Therefore, the value of a can be chosen to be any value

iii)

Hence, the function is

$f(x)=ax^2-8x-4$

Differentiate w.r.t x

$f'(x)=\frac{d}{dx}(ax^2-8x-4)=2ax-8$

This implies,

$f'(0)=2a(0)-8=-8$