Question

A quadratic polynomial with real coefficients and leading coefficient is called if the equation is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial for which the sum of the roots is maximized. What is

Answer 1

p(1) = 1

Let p(x) = x2 + bx = c

then, p(p(x)) = (x2 + bx + c)² + b(x2 + bx + c) + c

Sum of roots of p(p(x)) is given by = - (coefficient of x3/constant term)

= -2b/c2+bc+c = f(a,b) (say)

For critical points,

∂f/∂c = 0 and ∂f/∂b = 0

= 2b(2c+b+1)/c2+bc+c = 0 and - {( c2+bc+c)²- 2b (c)/ ( c2+bc+c)²}= 0

= b(2c+b+1) =0  

= b= 0 or b= - (1+2c)  = -(2c(c+1))/c2+bc+c = 0

If c=0 , b= -1  => c= 0 or c=-1

If c = -1 ,b = 1

thus we have the following possibility for (b,c)

(0,0) , (0,-1) , (-1,0) ,(1,-1).

At (0,0) f(0,0) = not defined

(0,-1) f(0,-1) = 0

(1,-1) f(1,-1) = 2

(-1,0) is not possible.

p(x) = (x2 +x-1)² + (x2 +x-1) -1

= p(1) = (1+1-1)² + (1+1-1) -1

= 1

Hence, p(1) = 1

Learn more about the sum of roots here brainly.com/question/10235172

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