Question

Let $f\left(x\right)$ is a polynomial function of degree four passing through the point $(0,1)$ and which increases in the intervals $(1,2)$ and $(3,\infty )$ and decreases in the interval $(-\infty ,1)$ and $(2,3).$ If $f\left(x\right)=0$ has $4$ distinct real roots, then the range of values of leading coefficient of the polynomial $f\left(x\right)$ is

• A. $[\frac{4}{9},\frac{1}{2}]$
• B. $(\frac{4}{9},\frac{1}{2})$
• C. $(\frac{4}{9},1)$
• D. $[\frac{1}{3},\frac{1}{2})$

Answer 1

The correct option is B $(\frac{4}{9},\frac{1}{2})$

From the given data we conclude that $\frac{df\left(x\right)}{dx}=0$ at $x=1,2,3$.
$\Rightarrow f^{\prime }\left(x\right)=a(x-1)(x-2)(x-3)$
$\Rightarrow f\left(x\right)=a\int (x^3-6x^2+11x-6)dx$
$=a(\frac{x^4}{4}-2x^3+\frac{11x^2}{2}-6x)+c$

Also, $f\left(0\right)=1\Rightarrow c=1$
$\Rightarrow f\left(x\right)=a(\frac{x^4}{4}-2x^3+\frac{11x^2}{2}-6x)+1$

$f\left(1\right)=1-\frac{9a}{4}=f\left(3\right)$
$f\left(2\right)=1-2a$


For roots should be real and distinct:
$f\left(2\right)>0$ and $f\left(1\right),f\left(3\right)<0$
$\Rightarrow 1-2a>0$ and $1-\frac{9a}{4}<0$
$\Rightarrow a<\frac{1}{2}$ and $a>\frac{4}{9}$
$\Rightarrow \frac{4}{9}<a<\frac{1}{2}$
$\Rightarrow a\in (\frac{4}{9},\frac{1}{2})$