Optimization (min/max and concavity)

[Angry Citizen]

This isn't a homework question, although I am in a calculus course. I'm a little fuzzy on the method that I was taught (discover intervals and all that nonsense to make sure f'(x)=0 is a max or a min). I was curious if, when I discovered the values of x such f'(x)=0, I could then find f''(x)=0 to determine if each f'(x) is a max/min, or merely a concavity point (thus, if f''(x)=0 is the same as f'(x)=0, it isn't a max/min).

Thanks!

 

[i2c]

f'(x) = 0 is for finding x values that you then plug back into f(x) to find mins and maxes.
you then check an x value on the left and the right of the f'(x) = 0 x value to determine if f(x) is increasing or decreasing on that interval. f''(x) = 0 is for finding x values that are inflection points. (where concave up or down changes to the other) again, you try values to the right and left of f''(x) = 0 to find the intervals that it's concave up and down

 

[Theorem.]

Yes you can use the second derivative test in optimization problems to verify that your x value is a maximum or minimum. if f is concave up (f''(x) is positive). you have a local minimum value. If it is concave down (f''(x) is negative) you have a local maximum. Although i would really recommend you take the time to learn the first derivative test with the intervals as it will often be easier than finding the second derivative. Additionally, the second derivative test fails when f''(x)=0 I have made an optimization tutorial on my website, please see the example and the first derivative test section. Here is a link: http://www.theoremsociety.com/forums/index.php?showtopic=6"

 

[Coriolis314]

Once you find the values of x such that f'(x)=0 you can pick arbitrary points around that value of x to determine if the point is a max/min/neither. You can also plug the value into f''(x); if the number that comes out is <0 you have a maximum, >0 minimum. Does that make sense?

 

[Theorem.]

Once you find the values of x such that f'(x)=0 you can pick arbitrary points around that value of x to determine if the point is a max/min/neither. You can also plug the value into f''(x); if the number that comes out is <0 you have a maximum, >0 minimum. Does that make sense?

I only apply this technique safely if the points represent the endpoints of a closed interval, following the extreme value theorem. If the endpoints are not clear or the interval is not closed, I generally use the first- or second derivative test (it is possible that there could be more that one local maxima and picking aribtrary points may not account for all of them- although this usually doesn't occur in elementary problems)