Possible max/mins on a cubic?

Question

I can see how differentiating works (setting the derivative = to 0). But apparently that only gives the "possible" points? How would I know with there is local extrema at those x-values? When would it be incorrect?

Answer 1

Yes, finding the points where the derivative=0 OR when it doesn't exists gives you some good idea of the "critical points".
But from then on it's much easier. You just need to find out whether the derivative is changing from positive to negative, or negative to positive. The reason why it is only "possible" is that if the Derivative changes from positive to 0, and then to positive again, you'll have a horizontal tangent, therefore there is no local extremum there. Same goes for negative to 0 to negative.
Don't forget that if the derivative doesn't exist, you can also have a local extremum. A corner, or a cusp can be created at that point, which are also local min and max's.
To tell if they are a local extremum, just find out if the derivative is positive/negative by plugging in some nearby numbers. it's that easy ;P. Positive to 0 to negative means a local max. Negative to 0 to positive means a local min.
For non-existent derivatives, first you have to find the left and right handed limits, find out if they're equal, THEN find out if it's changing from positive to negative or vice versa, and you're foudn out if it's a max or min ;)

Answer 2

You can use the second derivative test: If f"(a) < 0, then f(a) is a local max, and if f"(a) > 0, f(a) is a local min. If f" does not exist or is too difficult to compute, you can study the behavior of f': If f'(x) changes from <0 to = 0 to > 0 as x increases from < a to = a to > a, thenf(a) is a local min, and vice versa for a max.

It is possible for f'(a) to be 0, but f(a) is neither a max nor a min. For example, f(x) = x^3 has neither a max nor a min at x = 0, but f'(0) = 0.

Answer 3

Simplest way to tell max or min is to look at it graphically (if you're allowed). There are a couple of tests otherwise. 1st derivative test - if f' changes from - to + (from x-values just left to just right) then it's a min, if f' changes from + to -, then it's a max. 2nd deriv. test - f" > 0 implies max, f" < 0 implies min, f" = 0 inconclusive.
An inflection point on the graph has f' = 0 but neither max nor min. Typical example is y = x^3. It's graph has an inflection point at the origin where it levels out for a horiz tangent, although x^3 is increasing before and after x=0.

Answer 4

Check the second derivative
if it is less then 0 it's maximum
if it is geater then 0 it's minimum
if it is equal to 0 this method gives you nothing. try something else draw graph, etc