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 .
will generate an expression sequence of three roots and they can be referred to as solna,solna, and solna.
is of the form
Calculate the derivative of F.
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.