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