Tangent-plane invariants of special surfaces

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Problem 3552 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.5 "Applications in Geometry"

Prove that the tangent planes to the surface xyz=a3xyz=a^{3} (a>0)(a>0) form, with the coordinate planes, a tetrahedron of constant volume.

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Problem 3553 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.5 "Applications in Geometry"

Prove that the sum of the segments cut off on the coordinate axes by the tangent plane to the surface

x+y+z=a(a>0) \sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{a} \quad (a>0)

is a constant.

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Problem 3554 from Demidovich B.P. "Сборник задач и упражнений по математическому анализу" [Dem97]More problems in Section 6.5 "Applications in Geometry"

Prove that the tangent plane of the conical surface

z=xf(yx) z=xf\left(\dfrac {y}{x}\right)

passes through its vertex.

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Problem 3.6.17 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.6.2 "Problems"

Prove that the tangent planes to the surface

xyz=a3,a>0 x y z = a^{3}, \quad a > 0

form, with the coordinate planes, tetrahedra of constant volume. Find the volume of the tetrahedra.

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Problem 3.6.18 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.6.2 "Problems"

Prove that the tangent planes to the surface

x+y+z=a,a>0 \sqrt{x}+\sqrt{y}+\sqrt{z} = \sqrt{a}, \quad a > 0

cut off segments on the coordinate axes whose sum equals aa.

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Problem 3.6.19 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.6.2 "Problems"

Prove that the tangent planes to the surface

x2/3+y2/3+z2/3=a2/3,a>0, x^{2 / 3}+y^{2 / 3}+z^{2 / 3} = a^{2 / 3}, \quad a > 0,

cut off segments on the coordinate axes, the sum of whose squares is constant and equals a2a^{2}.

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Problem 3.6.21 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Функции многих переменных" [ККЧ+03]More problems in Subsection 3.6.2 "Problems"

Prove that all planes tangent to the surface

z=xf(y/x) z = x f(y / x)

where f(u)f(u) is a differentiable function, have a common point.

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References

[Dem97] Demidovich B.P. Сборник задач и упражнений по математическому анализу. 13-е изд., Издательство Московского университета / Издательство "ЧеРо", 1997.
[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Функции многих переменных. 2-е изд., ФИЗМАТЛИТ, 2003.