Find the absolute maximum and absolute minimum of f on the interval (-1,2]:
f(x)= (-x^3 + x^2 + 3x +1) / (x+1)
The Answer Choices:
a) Max: (1, -2) Min: (-1, 2)
b) Max: (1, -2) Min: None
c) Max: None Min: None
d) Max: None Min: (-1,2)
e) None of these
1) I found the derivative: ( -2x^3 - 2x^2 + 2x + 2 ) / (x+1)^2
2) I set the derivative equal to zero and solved for x, getting my critical points: x=1,x=2,x=3.
3) Since the interval is between (-1,2], I crossed out x=3.
4) I plugged in my critical points: x=1 and x= 2, and my end points: x=-1 and x=2 into f(x).
5) Results: f(-1)=undefined; f(1)=(2); f(2)=1.
So how do I determine which point is the max and which point is the min?
I tried my best at solving this myself;however I feel stuck after going this much through it. Could anyone review my work and tell me what I did wrong or what I need to do next in order to find which are my max and min points?
Thank you for any answers.
Think of it this way,
Now you can see that when x<1; dy/dx>0 That means that the graph of
f(x) when x<1 has a positive tangent. When x>1:dy/dx<0 that means that the graph of f(x) when x>1 has a negative tangent. So you see that clearly f(x) has a maximum at (1,2)
The second derivative test is also correct. But I think this is the easy way.
Please tell me whether this solves your problem.