Theory of Intelligence
11TH•Mathematics•Functions And Graphs•Question 5

Solution

Given $f(x)=x^3-ax^2+bx+1$, if $f(2)=-3$ and $f(-1)=0$, find the values of $a$ and $b$.

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Solution

Given:

f(x)=x3−ax2+bx+1f(x)=x^3-ax^2+bx+1

Use f(2)=−3f(2)=-3

f(2)=23−a(22)+b(2)+1=−38−4a+2b+1=−39−4a+2b=−3−4a+2b=−12−2a+b=−6(1)\begin{aligned} f(2)&=2^3-a(2^2)+b(2)+1=-3\\ 8-4a+2b+1&=-3\\ 9-4a+2b&=-3\\ -4a+2b&=-12\\ -2a+b&=-6\quad (1) \end{aligned}

Use f(−1)=0f(-1)=0

f(−1)=(−1)3−a(−1)2+b(−1)+1=0−1−a−b+1=0−a−b=0a+b=0(2)\begin{aligned} f(-1)&=(-1)^3-a(-1)^2+b(-1)+1=0\\ -1-a-b+1&=0\\ -a-b&=0\\ a+b&=0\quad (2) \end{aligned}

From (2): b=−ab=-a.

Substitute into (1):

−2a+(−a)=−6⇒−3a=−6⇒a=2-2a+(-a)=-6\Rightarrow -3a=-6\Rightarrow a=2

Then b=−2b=-2.

a=2,b=−2\boxed{a=2,\quad b=-2}

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