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", StyleBox["Mathematica", FontSlant->"Italic"], " uses the principal branch of the cube root. The cube root of a negative \ number is imaginary." }], "Text"], Cell[BoxData[ \(\((\(-1\))\)\^\(1/3\) // N\)], "Input"], Cell[BoxData[ \(f[1] // N\)], "Input"], Cell[TextData[{ "To take care of this problem, we change the way we define ", Cell[BoxData[ \(TraditionalForm\`f(x)\)]], "." }], "Text"], Cell[BoxData[ \(f[x_] := \((\((x - 2)\)\^2)\)\^\(1/3\)\)], "Input"], Cell["There are no stationary points.", "Text"], Cell[BoxData[ \(Solve[\(f'\)[x] == 0, x]\)], "Input"], Cell["Looking at the derivative,", "Text"], Cell[BoxData[ \(\(f'\)[x] // PowerExpand\)], "Input"], Cell[TextData[{ "we see that it is undefined at ", Cell[BoxData[ \(TraditionalForm\`x = 2\)]], "." }], "Text"], Cell[BoxData[ \(\(f'\)[2]\)], "Input"], Cell[TextData[{ "Since ", Cell[BoxData[ \(TraditionalForm\`f' \((x)\) < 0\)]], " for ", Cell[BoxData[ \(TraditionalForm\`x < 2\)]], " and ", Cell[BoxData[ \(TraditionalForm\`f' \((x)\) > 0\)]], " for ", Cell[BoxData[ \(TraditionalForm\`x > 2\)]], ", ", Cell[BoxData[ \(TraditionalForm\`x = 2\)]], " is a local minimum." }], "Text"], Cell[BoxData[ \(\(Plot[f[x], {x, 0, 4}];\)\)], "Input"] }, Open ]], Cell[CellGroupData[{ Cell["Cup of Minimum Surface Area", "Section"], Cell[TextData[{ "A cylindrical container with a circular base and an open top is to hold ", Cell[BoxData[ \(TraditionalForm\`64\ cm\^3\)]], ". 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