Answer:
No maximum; see explanation
Step-by-step explanation:
The easiest way to find the maximum value of this polynomial is to graph it. If you are not allowed to graph it, then the next easiest way to find the maximum would be to use calculus (if you can't use calculus I'll explain how to do that way as well).
Graphing:
Use a free graphing calculator online or a personal graphing calculator. You will see that this function has no maximum and continues upward forever. If you are interested in other methods to determine this they are below.
Using Calculus:
1. Find the derivative of the polynomial:
72x^3+22x+4
2. Set the derivative = 0 and solve for all solutions:
72x^3+22x+4 = 0
0 = 2(6x+1)(6x^2-x+2)
0 = (6x+1)(6x^2-x+2)
0 = (6x+1) Important equation 1
0 = (6x^2-x+2) Important equation 2
So x = -1/6 because 6x^2-x+2 has no solution
(This step produces critical points that you should test values near)
3. Test one point larger than -1/6 and one smaller than -1/6 by plugging them into the derivative:
Point 1: -2
Point 2: 0
72(-2)^3+22(-2)+4 = -616
72(0)^3+22(0)+4 = 4
Since the smaller term is negative and the larger term is positive x=-1/6 is a minimum value for this polynomial. Since only one x-value was found with the derivative test, this function has no maximum (it is technically infinity).
Without Calculus:
1. Set the polynomial = 0 and simplify:
18x^4+11x^2+4x+8 = 0
No simplification can happen.
(This step would give points that you should test values close to).
2. Test points
Plug points into the polynomial to get an understanding of the shape of the graph.
x=0 y=8;
x=1 y=41;
x=10 y=181148;
x=-100000000 (approximately negative infinity) y = approximately infinity
x=1000000000 (approximately positive infinity) y = approximately infinity
Based on the tested points, this graph has no maximum, it continues upward forever.
