PLZ BE CLEAR AND SHOW EVERYTHING THAT IS REQUIRED IN THE QUESTION THX The figure

Question

PLZ BE CLEAR AND SHOW EVERYTHING THAT IS REQUIRED IN THE QUESTION THX

The figure above shows a contour plot of the level curves of some function f(x, y). The solid lines show constant levels of positive while the dashed lines show constant levels of negative z. The dark, heavy lines shows level set z = 0. (a) At the points labeled A and B, clearly indicate the direction of the gradient vector nabla f. (b) Indicate all points where f(x, y) attains a local MAXIMUM. (c) Indicate all points where f(x, y) attains a local MINIMUM. (d) Indicate points where nabla f = (0, 0) but there is no local extreme of f(x, y).

Explanation / Answer

Hence, the directional derivative is the dot product of the gradient and the vector u. Now that if u is a unit vector in the x direction , u=<1,0,0>, then the directional derivative is simply the partial derivative with respect to x.

In mathematics, the gradient is a generalization of the usual concept of derivative to functions of seversal variables. If f(x1.....Xn) is a differentiable, real-valued function of several variables, its gradient is the vector whose components are the n partial derivatives of f.

The gradient vector is orthogonal to the level curve at the point . Likewise the gradient vector is orthogonal to the level surface at the point. Actually all we need here is the last part of this fact. This says that gradient vector is always orthogonal, or normal, too the surface at a point.

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