# (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).

Question-AnswerCategory: Mathematics(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) 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)