Handout 8 - Due Monday November 6

Problem 1
Let . When c=2, . This problem studies the behavior of solutions of as a function of the parameter c. Let be the function of c whose value is the smallest root of this polynomial. For example since -1 is the smallest root of .

a)

For each of and , find the smallest root of . Use these values to estimate when . (Note that the Maple

will generate an expression sequence of three roots and they can be referred to as solna[1],solna[2], and solna[3].

b)

Find the smallest roots for the values and , and use these to estimate when

c)

Find the smallest roots for the values and , and use these to estimate when

d)

Does the smallest root of appear to be a differentiable function of c when c=2? Explain !

e)

Show that for any numbers and , the cubic equation

is of the form

where

and

f)

Define by

Calculate the derivative of F.

g)

Show that the determinant of the derivative of F is zero when .

h)

Note that . Show using part g) that F does not have a differentiable local inverse G satisfying . Relate this to your answer in part d).

i)

Note that . Show that F does have a differentiable local inverse G satisfying .

Don't feel obliged to use the computer for each part above.

To ease the calculations, the above problem looked at the dependence of roots upon coefficients for cubic equations with no term. The (possible) local inverses G above are functions describing two of the roots as a function of the two nonzero coefficients. Part h) was looking at a typical case near where two roots were equal, while in part i), all roots were distinct.

By multiplying out , and defining an appropriate , one can similarly analyze the roots of a general cubic equation as a function of all three coefficients.