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I'm dealing with a specific polynomial function. The first derivative of it is displayed below. As you can see, it has the shape of an asymetric function. But what does this tells us about the initial function in general and in terms of finding a maximum or a minimum for it? I know it is a general question, but I find hardly any documentation on this and I think it's good if someone can give general information when the first derivative takes this form.

Any information is appreciated.

enter image description here

edit: when I have different inputs in the function, the first derivative looks like this:

enter image description here

Steven01123581321
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    This shape is not a parabola because it is clearly asymmetric. – Peter Foreman Oct 24 '19 at 14:54
  • It would have a shape similar to a negative cubic (not exactly the same, since its a bit asymmetric, and the gradient never hits $0$). If the derivative was actually a symmetric parabola, you could deduce the initial function was a cubic. – John Doe Oct 24 '19 at 14:56
  • @JohnDoe: it is hard to tell, but it looks to me like the derivative does hit $0$ at one point around $x=0.2$ to $0.25$ – Ross Millikan Oct 24 '19 at 14:57
  • @RossMillikan Ah yes, I just read your answer and realised that. – John Doe Oct 24 '19 at 14:59

2 Answers2

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It is not a parabola because a parabola is symmetric around the maximum. You can integrate it numerically to get the function up to an additive constant. The derivative looks like it hits zero about $0.2$ or $0.25$. Because the derivative is always negative except that one point, the function is always decreasing, rapidly when $x$ is away from $0.25$. It will have a "flat spot" like $y=x^3$ around $x=0.25$, then start decreasing rapidly again.

Ross Millikan
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HINT:

Value of the first derivative in $x$ is negative means that the original function is decreasing at $x$.

Value of the first derivative in $x$ is positive means that the original function is increasing at $x$.

Value of the first derivative in $x$ equals zero means that the original function has a stationary point at $x$ (minima, maxima or terrace point)

Vinyl_cape_jawa
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