Let f(x) be a quartic polynomial with distinct, real roots a,..., 04. (a) Show that S(x)= = f'(x) f(x) x-01 1 1 1 + + 2-02 2-03 2-04 and find a similar expression for T(x) = - d f'(x) dx f(x) (b) Leto and be real numbers such that oa, for j = 1,2,3,4 and Xxo, and consider the quadratic where K = === Q(x) = K(x-xo) - (x-1), (01-x) (02-A) (3-1) (as-A) + (0x0)2 (02-10) + - + (i) What can be said about roots of Q(a) between 0 and a;? (ii) By writing a;-= (a; - xo) + (xo - ), prove that K=4-2(xo-A)So + (xo - A)To: where So S(0) and To=T(xo). = (c) (i) Make the change of variables y=x-xo and =xo to obtain an expression for Q(y,) as a quadratic in . It turns out that the optimal choice for which moves the roots of Q as close as possible to the roots of foccurs at the point where a Denote this value by Ho: find its value, and then show that Q(y,Ho) = 0 if y 3T-S]-25oy-4=0. In your answer, you may assume that Toy 1. (ii) Is it possible to have Toy = 1 and aQ/a = 0 at the root? Why (or why not)? = 0. (d) By solving the quadratic from part (c) and writing fo= f(xo). fo= f'(xo) etc. show that 21=p(20) = 2 = 4fo fo [9f - 12o]/2 Hint: move the square root into the denominator before eliminating So and To-