The polynomial function is .

The polynomial has a relative minimum and relative maximum .

(a)

Find the degree of the polynomial.

The polynomial has two critical numbers.

So the degree of the function is atleast one greater than the number of critical numbers.

So the minimum degree of the polynomial function is .

(b)

So the third degree polynomial function is .

Apply derivative with respect to to find the critical numbers.

Equate it to zero to find the critical numbers.

The critical numbers are and .

Substitute the points and in the polynomial function.

(c)

Solve the Equations  and .

Substitute in .

Substitute , , and in the polynomial function.

(d)

Now verify the result using graphing utility.

(a) Minimum degree of the polynomial function is .

(b)

(c) .