Mathematics Stack Exchange
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Questions tagged [inflection-point]

For questions on inflection points, i.e., points on a smooth plane curve at which the curvature changes sign. Consider using with the [derivatives] tag if applicable.

In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (rarely inflexion) is a point on a smooth plane curve at which the curvature changes sign. In particular, in the case of the graph of a function, it is a point where the function changes from being concave (concave downward) to convex (concave upward), or vice versa.

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Can a function have a inflection point at somewhere non-differentiable?

Imagine this function : $$ f(x)=\begin{cases} x^2,\quad x<0 \\ \sqrt{x},\quad x\ge0. \end{cases} $$ In my lecture, my professor told me $f(x)$ is not differentiable at $0$ but $f(x)$ has an inflection point at $x=0$. I get the idea that…
BRAD ZAP
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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 particular, if $f$ has two local extrema $A$ and $B$,…
Joseph
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Inflection point characterization question

Define the mean of a function $f:\mathbb{R}\to\mathbb{R}$ of differentiability class $\mathcal C^2$ over $[a,b]$ by $$\operatorname{mean}_{[a,b]}(f) \;=\; \frac{1}{b - a} \displaystyle\int\limits_{a}^{b} f(x) \; \mathrm dx.$$ Fix a point…
Markus Klyver
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For all $h\in \mathbb{R}\sim\{0\}$, two distinct tangents be drawn from the point $(2+h,3h-1)$ to the curve $y=x^3-6x^2-a+bx$ then $\frac{a}{b}=?$

If for all $h\in \mathbb{R}\sim\{0\}$, two distinct tangents can be drawn from the points $(2+h,3h-1)$ to the curve $y=x^3-6x^2-a+bx$ then find value of $\frac{a}{b}$ My Attempt If two distinct tangents are to be drawn from point…
Maverick
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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 exists an open interval $I\subset (a, b)$, $x_0\in…
user791759
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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 the first part, do I have to check continuity of $f'$ in…
zaba12
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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 supposed to be 1.47? is it possible? I will use the…
mary jo
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inflection point with third derivative

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 definition of inflection points: Dfn: Let…
1123581321
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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 $x>-1$ and $f(x)=-1-x^{\frac{1}{3}}$ when $x<-1$. By…
a_i_r
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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 single line in $\mathbb{CP}^2$. I think I see that…
Ilcu_elte_smurf
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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 $C'$ (possibly reducible and singular) of degree $\ge 4$…
quantum
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