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Optimisation in several variables

Critical points, the second-derivative test, and Lagrange multipliers.

Critical points are where the gradient vanishes; the Hessian's determinant sorts them into minima, maxima and saddles. With a constraint, Lagrange multipliers say the gradients of f and of the constraint are parallel at the optimum. Picture it: the surface's bowl bottom, hilltop or saddle. The 3D view shows which. Think it: constrained optimisation is the geometry of tangency between level sets.

مثال عملي · gradient of x^2 + y^2 - 2x - 4y

Multivariable-Calculus · Partial-Derivatives

Gradient of x^2 - 2x + y^2 - 4y

x^{2} - 2 x + y^{2} - 4 y

خطوة خطوة

  1. f(x, y) = x^{2} - 2 x + y^{2} - 4 y

    The gradient is the vector of partial derivatives. Differentiate with respect to each variable, holding the others constant.

  2. \frac{\partial f}{\partial x} = 2 x - 2

    Treat every variable except x as a constant.

  3. \frac{\partial f}{\partial y} = 2 y - 4

    Treat every variable except y as a constant.

  4. \nabla f = \left[\begin{matrix}2 x - 2\\2 y - 4\end{matrix}\right]

    Assemble the gradient vector. It points in the direction of steepest ascent.

  5. (1, 2)

    Critical points: where every partial derivative is zero.

كشفت الإجابة
\nabla f = \left[\begin{matrix}2 x - 2\\2 y - 4\end{matrix}\right]

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كيف: Optimisation in several variables

  1. The gradient is the vector of partial derivatives. Differentiate with respect to each variable, holding the others constant.
  2. Treat every variable except x as a constant.
  3. Treat every variable except y as a constant.
  4. Assemble the gradient vector. It points in the direction of steepest ascent.
  5. Critical points: where every partial derivative is zero.

أسئلة يطرحها الناس

What is a partial derivative?

The ordinary derivative with respect to one variable while every other variable is frozen: the slope of the surface in one coordinate direction.

What does the gradient point at?

Uphill: the direction of steepest increase, with length equal to that steepest slope. It is perpendicular to the level curves.

أكثر في Multivariable Calculus

Functions of several variablesPartial derivatives and the gradientDouble and triple integralsVector fields, line integrals and the big theorems