This question is asking me to find the local max and min values of f using both the 1st and 2nd derivative tests. Then it asks me which method I prefer.
f(x)= x+ sqrt(1-x)
I don't have a problem with finding the derivative of the function, I already did that. I just have problems with the concepts of the 2 tests; they were not explained very well in my class.
The Attempt at a Solution
My understanding is that the 1st derivative test is the mean value theorum: where there exists a point c between point a and point b, and the derivative f ' (c) = f(b)-f(a)/ b-a.
For this 1st derivative test I would find critical points and then plug them back into the original equation to get extrema. Then use the Increasing/decreasing sign test to figure out if those values are relative minimum or relative maximum.
Please correct me if you have a better way of explaining the 1st derivative test.
I thought the 2nd derivative test required me to get the second derivative (f '' ), then set f '' = 0 and solve for x to get critical numbers. Then plug the critical numbers back into the original equation ( f) to get the points of inflection.
My problem is, it asks me which method I prefer. Which is confusing because I though that both of the tests gave you something different? The first test gives you relative maximum/minimum and the second test gives you the points of inflection. So is this trick question? You can't really prefer one because they give you different data.
I also wouldn't mind if you explained it to me using the function I provided, or if you didn't, I just need some help here.