Find the absolute maximum and absolute minimum of f on the interval (-1,2]: f(x)=(-x^3+x^2+3x+1)/(x+1)

A. Maximum: (1, -2); Minimum: (-1, 2)
B. Maximum: (1, -2); Minimum: None
C. Maximum: None; Minimum: None
D. Maximum: None; Minimum: (-1, 2)
E. None of these

I got to my minimum and maximum but when you plug in -1 to get the ordered pair it is undefined and I'm not sure what the answer would be.

f(x)=(-x^3+x^2+3x+1)/(x+1)

This factors and reduces to
f(x) = (x+1)(-x^2 + 2x + 1)/(x+1)
= -x^2 + 2x + 1 , x ≠ -1

f ' (x) = -2x +2 , for all x's except -1
-2x + 2 = 0
-2x = -2
x = 1 , which lies in your given domain of -1 to 2

so f(1) = -1 + 2 + 1 = 2
using the 2nd derivative test:
f ''(x) = -2, so we have a maximum

(1,2) is the only maximum point
looks like choice B

here is a graph of
f(x)=(-x^3+x^2+3x+1)/(x+1) from Wolfram
to verify my calculations.

http://www.wolframalpha.com/input/?i=plot+f%28x%29%3D%28-x%5E3%2Bx%5E2%2B3x%2B1%29%2F%28x%2B1%29+

To find the absolute maximum and absolute minimum of a function on a given interval, you need to follow a few steps:

1. Find the critical points by setting the derivative of the function equal to zero and solving for x.
2. Identify any endpoints of the interval.
3. Evaluate the function at the critical points and the endpoints.
4. Compare the function values to determine the absolute maximum and minimum.

Let's apply these steps to the given function f(x):

f(x) = (-x^3 + x^2 + 3x + 1)/(x + 1)

Step 1: Find the critical points by setting the derivative of f(x) equal to zero and solving for x.

To find the derivative, you can use the quotient rule:

f'(x) = [(x + 1)(-3x^2 + 2x + 3) - (-x^3 + x^2 + 3x + 1)(1)] / (x + 1)^2

Simplifying the derivative expression gives:

f'(x) = (5x^2 - x - 2) / (x + 1)^2

Setting f'(x) equal to zero:

5x^2 - x - 2 = 0

Factoring the quadratic equation gives:

(5x + 2)(x - 1) = 0

Solving for x, we get two critical points: x = -2/5 and x = 1.

Step 2: Identify the endpoints of the interval.

The given interval is (-1, 2], so the endpoints are -1 and 2.

Step 3: Evaluate the function at the critical points and endpoints.

Evaluate f(x) at x = -1, -2/5, 1, and 2:

f(-1) = (-1^3 + (-1)^2 + 3(-1) + 1) / (-1 + 1) = 0/0 (undefined)
f(-2/5) = [(-2/5)^3 + (-2/5)^2 + 3(-2/5) + 1] / (-2/5 + 1) = 41/21
f(1) = (1^3 + 1^2 + 3(1) + 1) / (1 + 1) = 3/2
f(2) = (2^3 + 2^2 + 3(2) + 1) / (2 + 1) = 7/3

Step 4: Compare the function values to determine the absolute maximum and minimum.

Comparing the function values, we find:

- Absolute maximum: (1, 3/2)
- Absolute minimum: (-2/5, 41/21)

Now, let's compare the results to the available answer choices:

A. Maximum: (1, -2); Minimum: (-1, 2)
B. Maximum: (1, -2); Minimum: None
C. Maximum: None; Minimum: None
D. Maximum: None; Minimum: (-1, 2)
E. None of these

Since the calculated points do not match any of the answer choices, it seems that none of the options provided are correct.