# Questions tagged [inflection-point]

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### Second derivative and inflection point

Let $f(x) = \begin{cases} \sin(\frac{1}{x})\cdot e^{-\frac{1}{x^2}}, & \text{if$x\neq0$} \\[2ex] 0, & \text{if$x=0$} \end{cases}$ Does $f''(0)$ exist? Is $x_0=0$ inflection point? Regarding ...
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### Critical and inflection points of the function $f(x)=|1+x^{\frac{1}{3}}|$

Determine the relative extrema and inflection points of $f(x)=|1+x^{\frac{1}{3}}|$. After breaking the modulus function we get a piecewise function such that $f(x)=1+x^{\frac{1}{3}}$ when $x>-1$ ...
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### Extrema of derivate are where tangent crosses the curve.

In this article https://www.jstor.org/stable/2310782 i found this proposition: Let $f$ be a differentiable function defined on an open interval $(a, b)$ containing the point $x_0$. Let: (B) There ...
• 350
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### how to find the inflection point of an exponential curve?

I have a set of decreasing numbers for example 18.98, 15.45, 11.7, 9.73, 9.06, 1.47, 0.1323, 0.1081, 0.0896, 0.0797, 0.0732. I want to find the inflection point in this set of numbers which is ...
• 75
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### Invariants of a quartic function

Some facts about cubic functions and quartic functions motivate this question: Every cubic function $f$ has exactly one inflection point $P$, and the graph $y=f(x)$ is symmetric about $P$. In ...
• 560
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### Why are all real inflection points on a cubic projective algebraic curve on 1 line?

Say we have $C\subset \mathbb{CP}^2$, a smooth curve of degree 3. I am aware of the group structure on cubics, what I don't get, is why are all inflection points with only real coordinates lie on a ...
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### quartic plane curves and their flexes

Suppose $C$ is a smooth, irreducible, quartic plane curve on the complex projective plane and let $P\in C$ be a flex (inflection point) on the curve. Is it true that any plane algebraic curve $C'$ (...
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Problem: Let $f:\mathbb{R}\to \mathbb{R}$ be a function such that $f'''(x)<0$ for every $x\not=0$ and $f''(0)=0$. Prove that $M(0,f(0))$ is an inflection point of $f$. I use the following ...