Determining Concavity of Functions over Their Domains · 确定定义域内函数的凹凸性
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| concavity/kənˈkævɪti/ | 凹凸性 | āo tū xìng |
| sign of the second derivative/saɪn ɒvðə ˈsekənd dɪˈrɪvətɪv/ | 二阶导数符号 | èr jiē dǎo shù fú hào |
| concave up/kɒnˈkeɪv ʌp/ | 上凹 | shàng āo |
| concave down/kɒnˈkeɪv daʊn/ | 下凹 | xià āo |
| point of inflection/pɔɪnt ɒv ɪnˈflekʃn/ | 拐点 | guǎi diǎn |
How the curve bends
- Increasing/decreasing tells you which way the curve goes; concavity 凹凸性 tells you how it bends.
- A curve can rise while bending like a cup (opening up) or like a cap (opening down).
- The bending is controlled by the sign of the second derivative 二阶导数符号.
- $f''$ measures how the slope $f'$ itself is changing.
曲线如何弯曲
- 递增/递减告诉你曲线朝哪个方向走;凹凸性告诉你它如何弯。
- 曲线可以一边上升,一边像杯子(开口向上)或像帽子(开口向下)那样弯。
- 弯曲由二阶导数符号控制。
- $f''$ 度量斜率 $f'$ 本身如何变化。
Concave up: $f'' > 0$
- Where $f''(x)>0$, the graph is concave up 上凹 — shaped like a smile, holding water.
- The slope $f'$ is increasing (getting more positive or less negative).
- Cup-shaped valleys are concave up.
- Tangent lines lie below a concave-up curve.
上凹:$f'' > 0$
- 在 $f''(x)>0$ 之处,图像上凹——像微笑,能盛住水。
- 斜率 $f'$ 在增大(更正,或更不负)。
- 杯形的谷是上凹。
- 切线在上凹曲线的下方。
A graph is concave up exactly where... · 图像恰好在其...
Concave up ⇔ $f''>0$ (slope increasing). · 凹向上 ⇔ $f''>0$（斜率增加）。
Concave up means the slope $f'$ is... · 凹向上意味着斜率$f'$是...
Concave up ⇔ $f'$ increasing ⇔ $f''>0$. · 凹向上 ⇔ $f'$递增 ⇔ $f''>0$。
Concave down: $f'' < 0$
- Where $f''(x)<0$, the graph is concave down 下凹 — shaped like a frown.
- The slope $f'$ is decreasing.
- Cap-shaped hills are concave down.
- Tangent lines lie above a concave-down curve.
下凹:$f'' < 0$
- 在 $f''(x)<0$ 之处,图像下凹——像皱眉。
- 斜率 $f'$ 在减小。
- 帽形的山是下凹。
- 切线在下凹曲线的上方。
Where · 何地 $f''(x)<0$, the graph is concave ____ (cap-shaped). · 当$f''(x)<0$时，图像是凹____（帽形）。
Negative second derivative → concave down. · 负二阶导数 → 凹向下。
Inflection points: where the bend flips
- A point of inflection 拐点 is where concavity changes — from up to down or down to up.
- There, $f''$ changes sign (usually passing through $0$, sometimes undefined).
- Build a sign chart of $f''$, just like you did for $f'$, to find these.
- An inflection point is about the bend changing, not the slope.
拐点:弯曲翻转之处
- 拐点是凹凸性改变之处——从上凹到下凹,或从下凹到上凹。
- 在那里,$f''$ 变号(通常经过 $0$,有时无定义)。
- 像对 $f'$ 那样,建一张 $f''$ 的符号表来找它们。
- 拐点说的是弯曲在变,而非斜率。
Concave down then up · 先凹向下再凹向上
y = x³
This cubic bends like a cap for $x<0$ and a cup for $x>0$ — the flip at $0$ is a point of inflection. · 这个三次函数在$x<0$处像帽子，在$x>0$处像杯子——在$0$处的翻转是拐点。
An inflection point requires $f''$ to change sign, not just equal zero. · 拐点要求$f''$变号，而不仅仅是等于零。
$y=x^4$ has $f''(0)=0$ but no sign change → not an inflection point. · $y=x^4$有$f''(0)=0$但没有变号 → 不是拐点。
For · 支持 $f(x)=x^3$, $f''(x)=6x$. At what $x$ is the inflection point? · 对于$f(x)=x^3$，$f''(x)=6x$。拐点在哪个$x$处？
$f''=6x$ changes sign at $x=0$. · $f''=6x$在$x=0$处变号。
For · 支持 $y=x^4$, is $x=0$ an inflection point? · 对于$y=x^4$，$x=0$是拐点吗？
$f''=12x^2\ge0$ on both sides — no sign change, so no inflection. · 两侧都是$f''=12x^2\ge0$——没有变号，所以没有拐点。
$f''(c)=0$ alone does not make $c$ an inflection point — $f''$ must actually change sign there. For $y=x^4$, $f''(0)=0$, but $f''\ge0$ on both sides (no sign change), so $x=0$ is not an inflection point. Check the sign on both sides, don't just solve $f''=0$.
仅有 $f''(c)=0$ 并不使 $c$ 成为拐点——$f''$ 必须在那里真正变号。对 $y=x^4$,$f''(0)=0$,但两侧 $f''\ge0$(不变号),所以 $x=0$ 不是拐点。检查两侧的符号,别只解 $f''=0$。
Find the concavity and inflection point of $f(x)=x^3$.
- $f'(x)=3x^2$, $f''(x)=6x$.
- $f''<0$ for $x<0$ (concave down); $f''>0$ for $x>0$ (concave up).
- $f''$ changes sign at $x=0$ → inflection point at $(0,0)$.
求 $f(x)=x^3$ 的凹凸性与拐点。
- $f'(x)=3x^2$,$f''(x)=6x$。
- $x<0$ 时 $f''<0$(下凹);$x>0$ 时 $f''>0$(上凹)。
- $f''$ 在 $x=0$ 变号 → 在 $(0,0)$ 有拐点。
Concavity is set by the sign of $f''$: $f''>0$ → concave up (cup), $f''<0$ → concave down (cap). A point of inflection is where $f''$ changes sign — the bend flips. $f''=0$ is a candidate, but you must confirm a sign change.
凹凸性由**$f''$ 的符号**决定:$f''>0$ → 上凹(杯),$f''<0$ → 下凹(帽)。拐点是 $f''$ 变号之处——弯曲翻转。$f''=0$ 是候选,但你必须确认变号。