Find the maximum of f(x, y, z)=x+y+z subject to the two constraints x^2+y^2+z^2=9 and (1)/(4) x^

Find the maximum of f(x, y, z)=x+y+z subject to the two...

Key Concepts

Quadratic Constraints

Quadratic constraints are those that involve quadratic expressions of the variables. Such constraints define geometrical shapes like circles, spheres, or ellipsoids in multiple dimensions, influencing the region over which the optimization is performed and generally adding a layer of complexity due to their non-linear nature.

Constrained Optimization

Constrained optimization involves finding the maximum or minimum of an objective function subject to equality or inequality constraints. In many real-life problems, variables cannot change freely due to these constraints, making this area essential in fields like economics, engineering, and applied mathematics.

Lagrange Multipliers

The method of Lagrange multipliers is a powerful technique for solving constrained optimization problems. By introducing auxiliary variables (multipliers) for each constraint, one sets up equations where the gradient of the objective function is equal to a linear combination of the gradients of the constraints, thereby converting a constrained problem into an unconstrained one.

Gradient Vectors

Gradient vectors indicate the direction of the steepest increase of a function. In the context of optimization, particularly with constraints, comparing the gradient of the objective function with the gradients of the constraints helps identify points where the function may achieve extrema under the given restrictions.

Transcript

00:01 We're given a function in two constraints, and we're asked to find a maximum of this function subject to these constraints.

00:09 The function is f of x, y, z equals x plus y plus z, and the constraints are sphere x squared plus y squared plus z squared equals nine, and the ellipse, one -fourth x squared plus one -fourth y squared plus four z squared equals nine.

00:38 And so we have the two constraint equations, g of x, y, z equals x squared plus y squared plus c squared equals 9.

00:54 And h of x, y, z equals 1 fourth x squared plus one fourth y squared plus 4 z squared equals nine.

01:13 So the, okay, we use the method of lagrange multipliers with two consistent.

01:23 To do this, we're out of the lagrange equations.

01:27 We have the gradient of f is the vector 1 -1 -1.

01:32 The gradient of g is the vector 2x, 2y, 2z, and the gradient of h is the vector x over 2, y over 2, 8 z, and the lagrange condition says that the gradient of f is equal to a linear combination of the gradients of a g and h.

02:00 So lambda gradient of g plus mu times the gradient of h.

02:05 And therefore we have that 1 -1 -1 is equal to lambda times 2x plus 2y, 2z, plus mu times x over 2, y over 2, 8z.

02:26 And so we obtain the three equations.

02:30 1 equals 2 lambda x plus 1 half mu x, 1 is equal to 2 lambda y plus 1 half mu y, and 1 is equal to 2 lambda y, and 1 is equal to 2 lambda z plus 8 mu z.

03:00 Now our first two equations will solve for lambda.

03:06 We know that x can't be 0 for these equations.

03:08 So we have that lambda is equal to 1 minus 1 half mu x over 2x, which simplifies to, well, okay, then we also have that from our second equation, lambda equals 1 minus 1 half mu y over 2y...

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