1. Left u(x, y) = e x sin y (a) Find the gradient of u 2. Let g(x, y) = x 2 + xy

Question

1. Left u(x, y) = ex sin y

(a) Find the gradient of u

2. Let g(x, y) = x2 + xy + 7y2 ? y3 ? 4x + 15

(a) One critical point of g is (1, 2). Verify this and determine which sort of critical point it is, using the discriminant.

(b) There is one other critical point. Either working directly from the formula for g, or by way of first making a substitution, find that other critical point and characterize it.

NOTE
The substitution you might want to make would be to say x = 1 + s, y = 2 + t. That converts g from its original formula in x and y into this: g = s2 + st + t2 ? t3. You would then find the critical point other than s = t = 0 and work back to the corresponding critical point for g in terms of x and y.

(The subsitution here is safe because partials and discriminants are not affected by translations, that is, substitutions of the form x = a + s, y = b + t. If you did something like go into polar, the discriminant
of the same quantity, given in terms of polar, could be different from what it would be in terms of Cartesian coordinates. Happily, local maxima, minima, and saddle points do not change character whenwe change from polar to Cartesian. But the details of the partials, and the value of the discriminant, typically do change.)

Explanation / Answer

1.

Consider the following function:

The gradient of is,

The unit vector in the direction of the vector is,

The directional derivative of the function in the direction at the point is,

2.

Consider the following function:

Find the partial derivatives:

At ;

The discriminant is,

Here, , then has local minimum at the critical point .

As per instructions, substitute in the given function.

Find the partial derivatives:

          ;

         ;

To find the critical points, equate and to zero.

Thus,

Therefore, the critical points other than is for .

Hence, the critical points for is .

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