The polynomial function is .

The polynomial has a relative minima , .

Relative maximum .

(a)

Find the degree of the polynomial.

The polynomial has three 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 .

Solve the system of equations using the calculator.

and

Substitute , , and in the polynomial function.

(d)

Now verify the result using graphing utility.

(a) Minimum degree of the polynomial function is .

(b)

(c) .