Question

The quadratic polynomials defined on real coefficients
$P\left(x\right)=a_1{x}^2+2b_{1}x+c_1$
$Q\left(x\right)=a_{12}{x}^2+2b_{2}x+c_2$ where $a_1\ne 0,a_2\ne 0$ and $P\left(x\right)$ and $Q\left(x\right)$ both take positive values $\forall \times \in Rg\left(x\right)=a_1{a}_2{x}^2+b_1{b}_{2}x+c_1{c}_2$ then

• A. $g\left(x\right)$ takes positive value only
• B. $g\left(x\right)$ takes negative value only
• C. $g\left(x\right)$ takes both positive and negative value
• D. nothing can be said about $g\left(x\right)$

Answer 1

The correct option is A $g\left(x\right)$ takes positive value only

$p\left(x\right)=a_1{x}^2+2b_{1}x+c_1$

$q\left(x\right)=a_2{x}^2+2b_{2}x+c_2$

both hold positive value for $\forall x\in R$

Then for $p\left(x\right);(2b_1{)}^2-4a_1{c}_1<0$ & $a_1>0$

$4b_1^2-4a_1{c}_1<0$ & $a_1>0$

$\Rightarrow b_1^2<a_1{c}_1$ .......(i)

for $q\left(x\right);(2b_2{)}^2-4a_2{c}_2<0$ & $a_2>0$

$4b_2^2-4a_2{c}_2<0$ & $a_2>0$

$b_2^2<a_2{c}_2$ ......(ii)

$g\left(x\right)=a_1{a}_2{x}^2+b_1{b}_{2}x+c_1{c}_2$

Here,

$(b_1{b}_2{)}^2-4a_1{a}_2{c}_1{c}_2$ .....(iii)

Since $b_1^2$ is $(+)$ve so $a_1{c}_1$ is also $(+)$ve

similarily $b_2^2$ is $(+)$ve so $a_2{c}_2$ is also $(+)$ve

From (i) & (ii) we get,

$b_1^2{b}_2^2<a_1{c}_1{a}_2{c}_2$

equation (iii) is;

$(b_1{b}_2{)}^2-4a_1{a}_2{c}_1{c}_2<0$ & $a_1{a}_2>0$

so, $g\left(x\right)$ is positive value for $\forall x\in R$.