Timeline for Intuition behind a zero derivative
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2023 at 22:41 | audit | Reopen votes | |||
| Dec 13, 2023 at 23:38 | |||||
| Dec 3, 2023 at 9:04 | audit | Reopen votes | |||
| Dec 3, 2023 at 9:12 | |||||
| Nov 20, 2023 at 21:40 | answer | added | DotCounter | timeline score: 0 | |
| Nov 20, 2023 at 18:09 | comment | added | Filip Milovanović | The tanget line is horizontal, but the value just left of the inflection point is slightly smaller, and just right of it is slightly larger (positive direction being left-to-right). | |
| Nov 20, 2023 at 16:34 | answer | added | hobbs | timeline score: 0 | |
| Nov 20, 2023 at 11:08 | comment | added | Ilmari Karonen | "I always thought that a zero derivative at a point $c$ means that for points $x$ really close to $c$, $f(x)=f(c)$." That's not even true for inflection points like $f(x)=x^2$ at $c=0$. A zero derivative just means that the tangent line is horizontal — nothing more, nothing less. What's mildly counterintuitive is that even a strictly increasing function can have a horizontal tangent line. | |
| Nov 20, 2023 at 5:13 | answer | added | AAM111 | timeline score: 3 | |
| Nov 20, 2023 at 1:26 | comment | added | Henry | If your function $f(x)$ is continuous at some $x=x_0$ and is increasing when $x<x_0$ and is increasing when $x>x_0$ then it must also be increasing at $x=x_0$ as there is nowhere else it can decrease or stay constant | |
| Nov 20, 2023 at 1:26 | answer | added | Cort Ammon | timeline score: 4 | |
| Nov 20, 2023 at 1:03 | history | became hot network question | |||
| Nov 19, 2023 at 21:23 | comment | added | Jean-Armand Moroni | Worse than that: a function can have its derivatives of all orders equal to zero, and still being strictly increasing. The well-known example is $\exp(-1/x^2)$ (for $x \ne 0$, and value $0$ on $0$): it is infinitely differentiable on $0$, and its derivatives of all orders are $0$ on $0$. | |
| Nov 19, 2023 at 18:41 | comment | added | John Douma | The derivative is only zero at a point. For a function to be constant in an interval around that point, all of the derivatives would have to be zero. | |
| Nov 19, 2023 at 17:09 | comment | added | Calvin Khor | "for x really close to c , f(x) = f(c)" means constant on a small interval around c. What is true is ""for x really close to c , f(x) is really close to f(c)", but that's just continuity. | |
| Nov 19, 2023 at 17:09 | answer | added | AnalysisStudent0414 | timeline score: 8 | |
| Nov 19, 2023 at 17:05 | history | edited | J. W. Tanner |
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| Nov 19, 2023 at 17:05 | comment | added | user2661923 | Draw the (simpler) graph of $~y = x^3,~$ whose derivative is $~= 0 ~$ at the isolated point of $~x = 0.~$ Everywhere else, the derivative is positive. The critical idea is that $~x=0~$ is an isolated point. | |
| Nov 19, 2023 at 17:00 | history | asked | Vivian | CC BY-SA 4.0 |