Here's a plot of a function:
Nothing special about this function – it's just $y = \sin{x}$. It's immediately clear, even without doing any calculations, which parts of the curve has positive first derivative, and which parts has positive second derivative. Parts of the curve that are increasing have positive first derivative, so e.g. between $-1$ and $+1$ the first derivative must be positive. Similarly if the function is concave up or down, it must have negative and positive second derivative respectively. So we can immediately say that between $0$ and $3$ the second derivative is negative.
How can I quickly tell where the 3rd and 4th derivatives are positive or negative? In this case it's easy to just do the math, but I'm looking to interpret what a result for "the fourth derivative is positive" or "the fourth derivative is more positive at $x = 3$ than at $x = 4$" says about the shape of the underlying function. Strictly, I am only interested in the fourth derivative, but I'm guessing that insight with the third derivative will help illustrate what happens with the fourth derivative.
Best Answer
No - it is not visually possible. The following chart is made up of parabolae with zero third and fourth derivatives almost everywhere, and looks almost identical to yours. One of them is $y=\frac{4x}{\pi}-\frac{4x^2}{\pi^2}$ and the others effectively repeat that one
We could go further and take double my approximation and then subtract your sine curve. This would have third and fourth derivatives opposite in sign to yours almost everywhere. On $[0,\pi]$ it would be $y=\frac{8x}{\pi}-\frac{8x^2}{\pi^2} -\sin(x)$
As a visual challenge, can tell which of the following is your sine curve and which has the opposite signs of third and fourth derivatives? (My initial suggestion is the black curve between them)