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Question:

Let for i=1,2,3, pi(x) be a polynomial of degree 2 in x, p'i(x) and p''i(x) be the first and second order derivatives of pi(x) respectively. Let , A(x) = ⎡⎢⎢⎣p1(x)p'1(x)p''1(x)p2(x)p'2(x)p''2(x)p3(x)p'3(x)p''3(x)⎤⎥⎥⎦ and B(x) = [A(x)]TA(x). Then determinant of B(x): Is a polynomial of degree 3 in x? Is a polynomial of degree 6 in x? Does not depend on x? Is a polynomial of degree 2 in x?

Does not depend on x

Is a polynomial of degree 6 in x

Is a polynomial of degree 3 in x

Is a polynomial of degree 2 in x

Solution:

pi(x) = aix2 + bix + ci ⇒ p'i(x) = 2aix + bi ⇒ p''i(x) = 2ai
B(x) = ⎡⎢⎣a1x2+b1x+c1a2x2+b2x+c2a3x2+b3x+c32a1x+b12a2x+b22a3x+b32a12a22a3⎤⎥⎦⎡⎢⎣a1x2+b1x+c12a1x+b12a1a2x2+b2x+c22a2x+b22a2a3x2+b3x+c32a3x+b32a3⎤⎥⎦
b11 = (a1x2+b1x+c1)2 + ((a2x2+b2x+c2)2 + (a3x2+b3x+c3)2
b22 = (2a1x+b1)2 + (2a2x+b2)2 + (2a3x+b3)2
b33 = 4a12 + 4a22 + 4a32
So, det.B(x) is a polynomial of degree 6 in x.