Question

Let $f\left(x\right)=x^4-4x^3+4x^2+c,c\in R.$ Then

• A. $f\left(x\right)$ has infinitely many zeros in $(1,2)$ for all $c$
• B. $f\left(x\right)$ has exactly one zero in $(1,2)$ if $–1<c<0$
• C. $f\left(x\right)$ has double zeros in $(1,2)$ if $–1<c<0$
• D. whatever be the value of $c,f\left(x\right)$ has no zero in $(1,2)$

Answer 1

The correct option is B $f\left(x\right)$ has exactly one zero in $(1,2)$ if $–1<c<0$

$f\left(x\right)=x^4-4x^3+4x^2+c$
$f^{\prime }\left(x\right)=4x^3-12x^2+8x$
$=4x(x-1)(x-2)$
Critical points are : $0,1,2$

For interval $(1,2)$,
$f\left(1\right)=1+c,f\left(2\right)=c$

$f\left(1\right)f\left(2\right)<0$
$\Rightarrow (1+c)c<0$
$\Rightarrow -1<c<0$
$\therefore f\left(x\right)$ has exactly one zero in $(1,2)$ if $–1<c<0$