Let f(x) be a quartic polynomial with distinct, real roots a,..., 04. (a) Show that S(x)=...

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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)² - (x-1)², (01-x)² (02-A)² (α3-1)² (as-A)² + (0x0)2 (02-10)² + - + (i) What can be said about roots of Q(a) between 0 and a;? (ii) By writing a;-λ= (a; - xo) + (xo - λ), prove that K=4-2(xo-A)So + (xo - A)²To: where So S(0) and To=T(xo). = (c) (i) Make the change of variables y=x-xo and μ=xo to obtain an expression for Q(y,) as a quadratic in μ. 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 y² 3T-S]-25oy-4=0. In your answer, you may assume that Toy² 1. (ii) Is it possible to have Toy² = 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± [9fő² - 12ƒoƒő]/2 Hint: move the square root into the denominator before eliminating So and To- 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)² - (x-1)², (01-x)² (02-A)² (α3-1)² (as-A)² + (0x0)2 (02-10)² + - + (i) What can be said about roots of Q(a) between 0 and a;? (ii) By writing a;-λ= (a; - xo) + (xo - λ), prove that K=4-2(xo-A)So + (xo - A)²To: where So S(0) and To=T(xo). = (c) (i) Make the change of variables y=x-xo and μ=xo to obtain an expression for Q(y,) as a quadratic in μ. 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 y² 3T-S]-25oy-4=0. In your answer, you may assume that Toy² 1. (ii) Is it possible to have Toy² = 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± [9fő² - 12ƒoƒő]/2 Hint: move the square root into the denominator before eliminating So and To-