Question

Find the absolute maximum and absolute minimum values of the function f on the given interval.

f(x)=ln(x^2+x+1) , [-1,1]

Answer 1

Find the critical numbers of the function. Then, plug the endpoints of the intervals and each critical number into the original function. Highest vaue is the absolute max, lowest value is the absolute min.

Answer 2

Critical numbers are found when you set the derivative equal to 0, then solve for x.

Answer 3

Ok, the derivative is 1/(x^2+x+1), right?

Answer 4

Also, will a critical value occur where the derivative doesn't exist?

Answer 5

Almost. You have to also multiply that by the derivative of what was inside the natural log.

Answer 6

Ok, forgot about the chain rule

Answer 7

Yup.

Answer 8

Hmm...I think a critical number can occur where the derivative doesn't exist, I can't really remember. But you're just concerned with the critical numbers between -1 and 1.

Answer 9

Yeah, and factoring the denominator provides a non-real answer, so it wouldn't matter anyway.

Answer 10

Alright, I think I've got it from here. Thanks for the quick help.

Answer 11

I guess there can't be a critical number if the derivative does not exist there....Still not sure lol. && No prob.

Answer 12

It says in my textbook that the definition of a critical number is:

A critical number of a function f is a number c in the domain of f such that either f'(c)=0 or f'(c) does not exist.

Answer 13

I think it means where a hole in the graph of the derivative occurs.

Answer 14

Undefined points that is