The polynomial function is .

The polynomial has a relative minimum at .

Relative maxima  and .

(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) .