(15%) Find the local extreme values and saddle point(s)(if any) of the function x2ex cos y, where x∈ R, −π 2 &lt

1. (15%) Find the local extreme values and saddle point(s)(if any) of the function x²e^{x cos y}, where x∈ R, −π

2 < y < 2π.

2. (10%) Find the tangent plane to x⁴+ y⁴+ z⁴= 9xyz at (1, 1, 2).

3. (15%) Find the absolute minimum value of the function√

x²+ 4y²on the curve xy= 2 by using Lagrange multiplier method.

4. (10%) Calculate I= ∫₋₁¹∫

1

−1(e^x2sin y+ x²y⁴)dxdy.

5. (15%) Evaluate I = ∬_T(x + y)¹⁰dxdy where T is the triangle with vertices(0, 0) , (1, 1) , (2, 0) 6. (10%) Compute the integrals I= ∫₀²∫

√4−y²

−√

4−y²x²y²dxdy.

7. (10%) Calculate I= ∫₀¹∫

1

√y

e^x3dxdy.

8. (15%) Let f(x, y) = ln√ x²+ y².

(a) Find the directional derivative of f at the point(3, 4) in the direction to point (2, 6).

(b) At what direction is the function f changing fastest at the point(3, 4)? What is the rate of change along this direction?

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