If the quadratic polynomials defined on real coefficient
`P(x)=a_(1)x^(2)+2b_(1)x+c_(1)` and `Q(x)=a_(2)x^(2)+2b_(2)x+c_(2)` take positive values `AA x in R`, what can we say for the trinomial `g(x)=a_(1)a_(2)x^(2)+b_(1)b_(2)x+c_(1)c_(2)` ?

Options

A. `g(x)` takes positive values only.
B. `g(x)` takes negative values only.
C. `g(x)` can takes positive as well as negative values.
D. Nothing definite can be said about `g(x)`.

Answer

A

Explanation

`(a)` `D_(1)=4b_(1)^(2)-4a_(1)c_(1) lt 0` and `a_(1) gt 0`
i.e. `a_(1)c_(1) gt b_(1)^(2)` ……..`(i)`
`D_(2)=4b(2)^(2)-4a_(2)c_(2) lt 0` and `a_(2) gt 0`
Hence `a_(2)c_(2) gt b_(2)^(2)` and `a_(2) gt 0`………`(ii)`
Multiplying `(i)` and `(ii)`, we get
`a_(1)a_(2)c_(1)c_(2) gt b_(1)^(2)b_(2)^(2)`
Now consider `g(x)`
`D=b_(1)^(2)b_(2)^(2)-4a_(1)a_(2)c_(1)c_(2) lt b_(1)^(2)b_(2)^(2)-4b_(1)^(2)b_(2)^(2)`
=-3b_(1)^(2)b_(2)^(2)`
`:. D lt 0`
`implies g(x) gt 0AA x in R (:' a_(1),a_(2) gt 0)`