Final answer:
The maximum value of the function g(x) = 1 + x + x^2 + x^3 + x^4 + x^5 on the interval [0, 2] is 63, and the minimum value is 1. This conclusion is based on the fact that g(x) has only positive coefficients, meaning it increases as x increases within the given interval.
Step-by-step explanation:
The maximum and minimum values of the function g(x) = 1 + x + x^2 + x^3 + x^4 + x^5 on the interval [0, 2] can be found by evaluating the function at the endpoints of the interval and at any critical points within the interval where the derivative is zero or undefined.
First, let's evaluate g(x) at the interval endpoints:
- g(0) = 1
- g(2) = 1 + 2 + 4 + 8 + 16 + 32 = 63
Next, we find the derivative of g(x):
g'(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4
Setting the derivative equal to zero to find critical points:
1 + 2x + 3x^2 + 4x^3 + 5x^4 = 0
This equation is difficult to solve analytically, so we would need a calculator or computer software to find approximate solutions for critical points.
However, since g(x) is a polynomial with only positive coefficients, g(x) will increase as x increases on the interval [0, 2]. Thus, the maximum value of g(x) will occur at the upper end of the interval, and the minimum value will occur at the lower end.
The maximum value of g(x) on [0, 2] is therefore 63, and the minimum value is 1.