maths.free › Calculus › Maxima and minima

Maxima and minima

Critical points and the second-derivative test.

A smooth function peaks or bottoms out only where its derivative is zero. Find those critical points, then use the second derivative to tell a maximum (f″ < 0) from a minimum (f″ > 0). This is the whole of optimisation in one variable.

例題 · critical points of x^3 - 3x

Calculus · Extrema

Critical points of x^3 - 3x

x^{3} - 3 x

ステップバイステップ

  1. f(x) = x^{3} - 3 x

    Critical points are where f′(x) = 0 or is undefined.

  2. f'(x) = 3 x^{2} - 3

    Differentiate.

  3. x = -1, x = 1

    Solve f′(x) = 0.

  4. f''(x) = 6 x

    Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

  5. f''(-1) = -6 \Rightarrow (-1, 2) \text{ is a local maximum}

  6. f''(1) = 6 \Rightarrow (1, -2) \text{ is a local minimum}

答えを表示
(-1, 2)\ \text{local maximum},\; (1, -2)\ \text{local minimum}

今度はあなたの番 問題を選択し、タイプしたり描いたりしてください。 すべてのステップは絵で、答えはあなたが尋ねるまで隠されています。

自分の解答を記録しよう

無料アカウントでは、すべてのレッスンにノートを追加し、完成したことの記録、解いた問題を一つの場所に保存し、このページについて質問できる先生を追加します。数学自体はログインしたかどうかに関係なく誰でも利用できます。

登録 ログイン

ここで使用された記号

どの記号をタップしても、定義、画像、それぞれの文字の意味が表示されます。

どうやって: Maxima and minima

  1. Critical points are where f′(x) = 0 or is undefined.
  2. Differentiate.
  3. Solve f′(x) = 0.
  4. Second-derivative test: f″ < 0 → maximum, f″ > 0 → minimum.

よくある質問

What is a derivative in one sentence?

The slope of the graph at a point, the rate at which the output is changing there. Speed is the derivative of position.

What is an integral in one sentence?

The accumulated total of a rate: the area under the curve. Distance travelled is the integral of speed.

Why are derivatives and integrals opposites?

That is the fundamental theorem of calculus: accumulating a rate and then measuring how fast the accumulation grows gets you back the rate. Integration undoes differentiation up to a constant.

When do I use substitution and when integration by parts?

Substitution when part of the integrand is the derivative of another part (u and du both present). Parts when the integrand is a product of two unrelated kinds of function, a polynomial times an exponential, log or trig function.

More in Calculus

LimitsDerivativesIntegralsDefinite integralsTaylor seriesSeries and sumsThe chain ruleImplicit differentiationRelated rates and optimisationIntegration techniques: substitution, parts, partial fractionsApplications of integration: area, volume, arc lengthInfinite series and convergence tests