Let f(x)=x²+bx+c,where b,c belong to R.If f(x) is a factor of both x⁴+6x²+25 and 3x⁴+4x²+28x+5,then the least value of f(x) is

^a.2

^b.3

^c.5/2

^d.4

Hi Menka,

Concept:

Herer the idea is to get a Quadratic Equation (so that you can find its minimum)

Now if f(x) is a factor of P(x) and Q(x), then f(x) should be a factor of any linear combination of P(x) and Q(x)-------- {this is obvious because, f(x) is indivdually a factor for both the functions}

Hence f(x) is factor of 3P(x) - Q(x) = 14x^2 - 28x + 70 = 14{x^2 - 2x + 5}

Now as f(x) is Quad, with co-eff od x^2 as 1, f(x) must be x^2 - 2x + 5.

ie f(x) = (x-1)^2 + 4.

So minimum value is 4.

Option (D). Hope that helps.

Best Regards,

Ashwin (IIT Madras).