Question

How do you use the first and double derivative to find the maxima and minima of a function

Answer 1

Say that we have some differentiable function f(x). We can find potential candidate(s) for local extrema by solving f ′(x)=0 for x.

We can now use the second derivative to determine the nature of this candidate, say x0x0.

If f ′′(x0)>0, then x0 is a local minimum, which may or may not be a global minimum.

If f ′′(x0)<0, then x0 is a local maximum, which may or may not be a global maximum.

However, if f ′′(x0)=0, then the test is inconclusive, x0 may be an inflection point or a maxima/minima. Similarly if f ″(x0)is undefined