Question

Let a,b,c be the three sides of a triangle and $f\left(x\right)=b^2{x}^2+(b^2+c^2-a^{2}x+c^2)$ then which of the following is /are CORRECT?

• A. $f\left(x\right)>0\forall xϵR$
• B. $f\left(x\right)>0<xϵR$
• C. $f\left(1\right)>0$
• D. f(a+b+c)=0

Answer 1

Answer: (A)

$f\left(x\right)>0\forall xϵR$

We have,

$f\left(x\right)=b^2{x}^2+(b^2+c^2-a^{2}x+c^2)$

$f\left(x\right)=b^2{x}^2-a^{2}x+b^2+2c^2$

Comparing that,

$f\left(x\right)=A{x}^2+Bx+C$

$A=b^2,B=-a^2,C=b^2+2c^2$

Then, We know that,

$D=B^2-4AC$

$={(-a^2)}^2-4b^2(b^2+2c^2)$

$=a^4-4b^4-8b^2{c}^2$

Hence, $a,bandc$ are constant term

Which is also greater than zero.

So, $f\left(x\right)>0\forall x\in R$

Hence, this is the answer.