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# Multiple Variable Calculus

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Question 1.

Find the critical points of the function

f(x, y) = 4xy2 - 4x2 - 2y2

and classify them as local maxima, minima or saddles or none of these.

Question 2.

The surface is defined by

z = 3x2 + 2y2 - 3

Find the equation of the tangent plane to the surface at the point (1, 1, 2).

Question 3.

(a) Find the directional derivative of the function f(x, y) = x2ey at the point (1,0) in the direction i + 2j.
(b) Find the gradient vector of f (x, y). In which direction, (i + 2j) or (2i + j), does f(x, y) change more rapidly? Explain.

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#### Solution Preview

Question 1.

Find the critical points of the function

f(x, y) = 4xy2 - 4x2 - 2y2

and classify them as local maxima, minima or saddles or none of these.

Solution:

To find stationary points, put = 0 and 0
This gives
= 0-----------------(ii)

Put y = 0 in (i)

Put x=1/2 in (i)

Critical points are (0,0), (1/2,-1) and (1/2, 1).

D = fxxfyy - fxy2

D = -8(8x-4) -(8y)2 = -64x + 32 - 64y2

D(0,0) =-64(0) + ...

#### Solution Summary

This solution has detailed explanations of questions on finding local maxima, minima or saddle points of functions of two variables, directional derivative and gradient vector.

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