#The derivative tests Both of the derivative tests essentially do the same thing, test whether or not a critical point is a local minima or a local maxima, you should still know both of them because if one gives an undefined value you will need to use the other in order to obtain a solution. First of, lets look at the first derivative test. ##The first derivative test If \(c\) is a critical point of a function \(f\) and \(f\) is continuous at the point \(c\) and is differentiable on an open interval containing \(c\) then the following are true: 1. If \(f'\) changes from positive to negative at \(c\) then \(f(c)\) is a local maximum of \(f\). 2. If \(f'\) changes from negative to positive at \(c\) then \(f(c)\) is a local minimum of \(f\). 3. If \(f'(x) \lt 0\) or if \(f'(x) \gt 0\) for every \(x\) in the open interval which contains \(c\) except at \(x = c\) then \(f(c)\) is not a extremum of \(f\). ##The second derivative test If \(c\) is a critical point of \(f\) and \(f\) is differentiable on an interval containing \(c\) then the following are true: 1. If \(f''(c) \gt 0\) then \(f\) has a local minimum at \(c\) 2. If \(f''(c) \lt 0\) then \(f\) has a local maximum at \(c\) ##Conclusion I personally prefer the second derivative test, but you may not as it employs more work when finding the derivatives as you have to take two derivatives rather than one. However, you have to use the first derivative instead of the second derivative test if \(f''(c) = 0\).