GeoGebraで増減表を作成するオリジナルツール「DerivativeChart」

使用例

ダウンロード

DerivativeChart Tool - GeoGebra

説明

指定した区間における、関数の増減表を返します。同時に、指定した区間における極大点、極小点、変曲点のリストを返します。

ただし、関数は、指定区間において微分可能でなければ、正しい結果を返しません。

構文は、

DerivativeChart[ <xの関数>, <xの開始値>, <xの終了値> ]

です。

このツールは、GeoGebra バージョン 6.0.518.0-w (20 December 2018) にて動作確認しました。本ツールはβ版です。

バグ情報（随時修正予定）

・引数に1次関数を指定すると、f''(x)の判定や、増減の矢印の表示に問題あり。すなわち、区間内で、f''(x)が一定値0をとる場合（増減の矢印がまっすぐな場合）に未対応。

（参考）

DerivativeChart[ f, min, max ]の戻り値は、以下のように定義されています。

極大点・極小点

Zip[( x(s), f( x(s) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]]

変曲点

Zip[( x(s), f( x(s) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]

増減表

TableText[{Append["x", Sort[Join[{Zip["" + (t), t, Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]], Zip["•••", s, Sequence[(Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t] + Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t + 1]) / 2, t, 1, Length[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]] - 1]]}], Join[{Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], Sequence[(Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t] + Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t + 1]) / 2, t, 1, Length[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]] - 1]}]]], Append["f'(x)", Sort[Join[{Zip[If[y(s) > 0, "+", y(s) < 0, "-", y(s) ≟ 0, "0"], s, Unique[Sort[Flatten[{(x( ( min, f(min ) ) ), f'(x( ( min, f(min ) ) ) ) ), (x( ( max, f(max ) ) ), f'(x( ( max, f(max ) ) ) ) ), Zip[(x(t), 0), t, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(t), f'(x(t ) ) ), t, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], Zip[If[f'(s) > 0, "+", "-"], s, Sequence[(Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t] + Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t + 1]) / 2, t, 1, Length[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]] - 1]]}], Join[{Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], Sequence[(Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t] + Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t + 1]) / 2, t, 1, Length[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]] - 1]}]]], Append["f''(x)", Sort[Join[{Zip[If[y(s) > 0, "+", y(s) < 0, "-", y(s) ≟ 0, "0"], s, Unique[Sort[Flatten[{(x( ( min, f(min ) ) ), f''(x( ( min, f(min ) ) ) ) ), (x( ( max, f(max ) ) ), f''(x( ( max, f(max ) ) ) ) ), Zip[(x(t), f''(x(t ) ) ), t, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(t), 0), t, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], Zip[If[f''(s) > 0, "+", "-"], s, Sequence[(Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t] + Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t + 1]) / 2, t, 1, Length[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]] - 1]]}], Join[{Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], Sequence[(Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t] + Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t + 1]) / 2, t, 1, Length[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]] - 1]}]]], Append["f(x)", Sort[Join[{Zip["" + (s), s, Zip[y(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]], Zip[If[(f'(s) > 0) ∧ (f''(s) > 0), "⤴", (f'(s) > 0) ∧ (f''(s) < 0), "↱", (f'(s) < 0) ∧ (f''(s) > 0), "↳", (f'(s) < 0) ∧ (f''(s) < 0), "⤵"], s, Sequence[(Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t] + Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t + 1]) / 2, t, 1, Length[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]] - 1]]}], Join[{Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], Sequence[(Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t] + Element[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]], t + 1]) / 2, t, 1, Length[Zip[x(s), s, Unique[Sort[Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}], Zip[x(s), s, Flatten[{(min, f(min ) ), (max, f(max ) ), Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f'(x), x - x, min, max]}]], Zip[(x(s), f(x(s ) ) ), s, RemoveUndefined[{Intersect[f''(x), x - x, min, max]}]]}]]]]]] - 1]}]]]}, "c_|h"]