What does it tells us if first derivative is a parabola?
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.
edit: when I have different inputs in the function, the first derivative looks like this:
2 Answers
$\begingroup$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.
$\endgroup$ $\begingroup$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)
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