Question

For the quadratic polynomial $f(x)=4x^2-8kx+k$, 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 $k(1+12k)$

Answer 1

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

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

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

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

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

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

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

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

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