Question

Find the extreme values of\(f\)on the region described by the inequality.

\(f(x,y) = 2{x^2} + 3{y^2} - 4x - 5,\;\;\;{x^2} + {y^2} \le 16\)

Short answer

Maximum value is\(47\)and minimum value is\( - 7\).

Step-by-step solution

Method of Lagrange multipliers

To find the maximum and minimum values of\(f(x,y,z)\)subject to the constraint\(g(x,y,z) = k\)(assuming that these extreme values exist and\(\nabla g \ne {\bf{0}}\)on the surface\(g(x,y,z) = k)\):

(a) Find all values of\(x,y,z\), and\(\lambda \)such that\(\nabla f(x,y,z) = \lambda \nabla g(x,y,z)g(x,y,z) = k\)

And

(b) Evaluate\(f\)at all the points\((x,y,z)\)that result from step (a). The largest of these values is the maximum value of\(f\); the smallest is the minimum value of\(f\).

Find critical point

Consider\(f(x,y) = 2{x^2} + 3{y^2} - 4x - 5\)

\(\begin{array}{l}{f_x} = 4x - 4\\{f_y} = 6y\end{array}\)

Critical points\({f_x} = 0,{f_y} = 0\)

\(\begin{array}{c}4x - 4 = 0\\x = 1\end{array}\)

And

\(\begin{array}{c}6y = 0\\y = 0\end{array}\)

The critical points is\((1,0)\)

Evaluate\(\nabla f = \lambda \nabla g\)

Consider\(f(x,y) = 2{x^2} + 3{y^2} - 4x - 5\)and.

\(\nabla f = \lambda \nabla g\)

\(\begin{array}{c}\nabla \left( {2{x^2} + 3{y^2} - 4x - 5} \right) = \lambda \nabla \left( {{x^2} + {y^2}} \right)\\4x + 6y - 4 = \lambda \left( {2x + 2y} \right)\end{array}\)

Comparing both sides:

\(\begin{array}{c}4x - 4 = \lambda 2x \cdots (1)\\6y = \lambda 2y \cdots (2)\\{x^2} + {y^2} = 16 \ldots (3)\end{array}\)

Find extreme points

If\(y = 0\), then from equation (3) \(x = \pm 4\)

If\(y \ne 0\), then from equation (2) \(\lambda = 3\)

Using this in equation (1) we get \(x = - 2\)

And then from (3) \(y = \pm 2\sqrt 3 \)

Then all the extreme points are\((4,0),( - 4,0),( - 2,2\sqrt 3 ),( - 2, - 2\sqrt 3 )\).

Find maximum & minimum value

Compute function at the critical point with the extreme values on the boundary

\(\begin{array}{c}f(1,0) = - 7\\f(4,0) = 11\\f( - 4,0) = 43\\f( - 2,2\sqrt 3 ) = 47\\f( - 2, - 2\sqrt 3 ) = 47\end{array}\)

Therefore the maximum value is\(47\).

And the minimum value is\( - 7\).