Question

Consider $f(x)={\left(x^2-3x+2\right)}^2-2\left(x^2-x-2\right)(x-1{)}^2+3{\left(x^2-1\right)}^2,x\in R$. Identify which of the following statement(s) is / are correct?

• A. Equation $f(x)=0$ has no solution.
• B. Equation $f(x)=0$ has exactly one solution.
• C. Least integral value of $\frac{f(x)}{{\left(x^2-1\right)}^2}$ is 3 .
• D. Least integral value of $\frac{f(x)}{{\left(x^2-1\right)}^2}$ is 2 .

Answer 1

Answer: (B), (C)

$f(x)=(x-1{)}^2(x-2{)}^2-2(x-2)(x+1)(x-1{)}^2+3(x-1{)}^2(x+1{)}^2$
$f(1)=0\Rightarrow x=1$ is the solution of the equation $f(x)=0$
$\frac{f(x)}{{\left(x^2-1\right)}^2}={\left(\frac{x-2}{x+1}\right)}^2-2\left(\frac{x-2}{x+1}\right)+3$
$g(t)=t^2-2t+3,\text{ where }t=\frac{x-2}{x+1},t\ne 1,-\frac{1}{2}$
$=(t-1{)}^2+2$
$\therefore g(t)>2$