4th post

Well this was a very interesting week i must say. Learning how to take the first and second derivative proved to be a challenge to me at first, but i believe i sort of understand how to do it now. If the problem asks you to find the intervals on which the function is increasing or decreasing then you take the first derivative, then set that equal to zero and then solve for x. For example on the homework problem #3, the equation is x^2-6x+8. First you would take the derivative on that, which would be 2x-6. Then you would set that equal to zero 2x-6=0. and then solve for x. which would give you x=3. The only thing i do not understand is what to do after that. Do i plug in numbers from the graph to give me the intervals or do i use the (-infinity, 3) u (3, infinity) as my intervals?

Also, when the homework asks for critical points. I know you have to take the first derivative and then solve for x again. But if x=0, how do u find other critical points or is zero your critical points? I know how to do the easy ones, for example,

x^2-6x, they asks for the critical points and all other possible extrema. So you take the derivative which is 2x-6. Then set equal to zero, then solve for x. Which will give you x=3. So my critical point is 3. Which will give me the intervals (-infinity,3) u (3,infinity). after that we take numbers in between those points to plug into our derivative to find out whether the graph is increasing or decreasing. It will also tell us if the numbers are max's or min's. That part i understand, the number one thing i do not understand is how to justify my answers. For that problem i would say "by using the first derivative test, i took the derivative of the original function and set the derivative equal to zero. Then i solved the function for x. When i found the x point i set up intervals. Then i plugged numbers in between my intervals into the derivative i found using the first derivative test. By plugging these numbers in i found that the first interval was decreasing and the second interval was increasing. I then found that the critical point 3 is a min." did i justify that correctly or did i do something wrong in the problem. I think i did all the tests right but i'm not sure. Can someone help me on my justifications and how to find the critical points and extrema. Thanks :)