Why is the second derivative test inconclusive for some local max/mins? I know what the second derivative test is, when it is can be used, and when it can't.  So I am not asking any of those questions.  What I am asking is why we could have a local max at $c$, have $f'(c)=0$ yet have $f''(c)=0$.  I am looking for a graphical answer.  For instance, $x^4(x-1)^3$ has a local max at $x=0$.  However, $f''(0)=0$.  Whether I look at the definition of concave down which says the tangent lines around 0 should be above the graph or if I think of it as opening down at $x=0$, I feel like $f''(0)$ should be negative but it is not.
 A: If the derivative of a function is zero, locally the only thing we do know is that the function has stopped increasing or decreasing at that moment. Similarly, if the second derivative is zero, it means the first derivative has stopped increasing or decreasing. With only this knowledge, is not known what will happen 'after', or what happened 'before'.
This will happen in the following scenario: if a function has a root of multiplicity at least 3 at $x=a$, then $f(a)=f'(a)=f''(a)=0$.
However: IF $f''(a)> 0$. We know that the function was accelerating "upward" before and after $x=a$ so in contrast to the other case, we do know some behavior of the function. If $f'(a)=0$ as well, it has stopped increasing or decreasing, and the function has been accelerating in the positive direction. Obviously, it hasn't been increasing, and stopped due to the positive acceleration, so it must have been decreasing before $x=a$ and increasing after $x=a$.
A: Just think about really simple polynomials - consider $f(x) = x^3$ and $f(x) = x^4$. It should come as no surprise that one of these has a local minimum at $0$ and the other does not - when the linear and quadratic terms of a polynomial are zero, the local behaviour is determined by the higher-order terms. If the leading term is cubic then you have an inflection point, while if it is quartic you have a local extremum.
Taylor series provide the intuitive way to apply this knowledge about polynomials to more general functions. In general the local behaviour of an analytic function is determined by its lowest-order non-zero derivatives - if the first $n$ derivatives are zero, then the function will locally look like the polynomial $x^{n+1}$. Thus if $f'(c) = 0$ and $f''(c) = 0$ then you need to move to a "fourth derivative test" - you can conclude a local minimum if you have $f^{(3)}(c) = 0$ and $f^{(4)}(c) \ge 0$.
