** 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.