Question

For the quadratic polynomial $f(x)=4 x^2-8 k x+k$, the statements which hold good are

• A. there is only one integral $k$ for which $f(x)$ is non negative $\forall x \in R$
• B. for $k< 0$ the number zero lies between the zeros of the polynomial.
• C. $f( x )=0$ has two distinct solutions in $(0,1)$ for $k \in(1 / 4,4 / 7)$
• D. Minimum value of $y \forall k \in R$ is $k (1+12 k )$

Answer 1

Answer: (A), (B), (C)

(A) $ f ( x )$ is non negative $\forall x \in R$
$\Rightarrow f ( x ) \geq 0 \forall x \in R \Rightarrow D \leq 0 $
$ 64 k ^2-16 k \leq 0 $
$ 4 k ^2- k \leq 0 $
$ k (4 k -1) \leq 0$

$\therefore $ integral value of $k =0$
(B) $f (0)<0$
$k <0 \Rightarrow (B)$

(C) for distinct roots in $(0,1)$

$D >0 \Rightarrow k (4 k -1)>0 $ ....(1)

$0<-\frac{ b }{2 a }<1 \Rightarrow 0< k <1 $ ...(2)

$f (0)>0 \Rightarrow k >0$ ....(3)

$f (1)>0 \Rightarrow 4-7 k >0 $
$\Rightarrow k < 4 / 7$ .....(4)

$(1) \cap(2) \cap(3) \cap(4) \Rightarrow k \in(1 / 4,4 / 7)$