(a) Find the maximum and minimum values of the function f(x, y, z) = 3x + 6y + 2z subject to the constraint 2x^2+4y^2+z^2=70. (b) Consider f (x, y) = xye^x-y (i) Find the rate of change of the function f(x,y) at the point P(5,5) in the direction of the vector u=(1,2). (ii) In which direction does the functionf(x, y) increase most rapidly at the point P(5,5)? (iii) What is the maximum rate of change off(x,y) at the point P(5,5)? (c ) Let f(x, y) = x^2 + 2y^2 – x^2y + 5. Find all critical points of f(x,y) and determine their nature (local max or local min or saddle point).

(a)

subject to the constraint:

Using Lagrange Multiplier , we get the following equations by Lagrange Multiplier Method:

which give the following relations for x, y, z:

Plugging these values in the constraint:

So, the respective values of x,y,z are:

So, we have two critical points: (3,3,4) and (-3,-3,-4). So,

So, the **maximum value of the function f(x,y,z) is 35** and **the minimum value of the function is -35.**

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(b)

(i)

So, at (5,5)

The unit vector for u(1,2) is:

So, the directional derivative for f(x,y) in the direction of u is:

(ii)

The function increases most rapidly in the direction of its gradient.

At point (5,5):

So, at (5,5), the function increases most rapidly in the direction <30 , -20>.

(iii)

The maximum rate of change is the absolute value of the gradient at that point, ie.,:

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(c)

The criticcal points will be when both equations are simultaneously zero.

So, critical points are:

The double derivatives are:

So, the determinant is calulated as:

Since both values are greater than 0, **f(x,y) has relative minima at (0,0).**

Since D < 0 , **f(x,y) has saddle point at (2,1)**

**Since D < 0, f(x,y) has saddle point at (-2,1)**