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9. Please find the solutions to this question. Thanks!!

Mathematics
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Apr 7th, 2015

a) The graph behaves like x^4 for large values of |x|.

b) x^4 - 6x^3 + 9x^2 - 4x + 12 = (x-3)(x^3 - 3x^2 + 4) = (x-3)(x-1)(x^2 - 4x - 4) = 0

at x = 1, 3, 2 + 2sqrt{2}, 2 - 2sqrt{2} - these are x-intercepts.

Possible rational roots: answer B (the leading coefficient is 1, the constant coefficient is 12, has factors 1, 2, 3, 4, 6, and 12. Together with their opposites, the factors constitute all the possible rational roots of the function.

The factored form (x - 3) (x - 1) (x - 2 - 2sqrt{2})(x - 2 + 2sqrt{2}) (or (x-3)(x-1)(x^2 - 4x - 4)  if we do not use irrational numbers).

The graph crosses the x-axis at the points (1, 0),  (3, 0), ( 2 + 2sqrt{2},  0), (2 - 2sqrt{2}, 0).

The graph never touches   the x-axis because all the roots are simple (multiplicity 1).



Apr 6th, 2015

What are the maximum number of turning points?

Apr 6th, 2015

Since the degree of the polynomial f(x) equals 4, the maximum number of turning points is one less, that is, 3.

Actually, between any two roots of the function there is a turning point and the function has four roots                (1, 2  - 2 sqrt{2}, 3, 2 + 2 sqrt{2}), so the number of turning points is exactly 3. 

Apr 7th, 2015

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