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Let f(x)=x 2 +bx+c,where b,c belong to R.If f(x) is a factor of both x 4 +6x 2 +25 and 3x 4 +4x 2 +28x+5,then the least value of f(x) is a.2 b.3 c.5/2 d.4

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 

Grade:12th Pass

1 Answers

Ashwin Muralidharan IIT Madras
290 Points
10 years ago

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).

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