Final answer:
Using Lagrange multipliers, we can find the highest point on the curve of intersection by considering the plane equation as a constraint and solving the resulting system of equations, then checking for the maximum z-coordinate value.
Step-by-step explanation:
To find the highest point on the curve of intersection of the given surfaces using Lagrange multipliers, first identify the surfaces involved: a cone given by x²+y²−z²=0 and a plane given by x+2z=4. We treat the equation of the plane as a constraint and the equation of the cone as the function to maximize or minimize.
To apply the method of Lagrange multipliers, set up the function L(x, y, z, λ) = f(x, y, z) + λ(g(x, y, z)), where f(x, y, z) is the function we wish to optimize, and g(x, y, z) = 0 is the constraint function.
Next, take the gradient of L and set it equal to the zero vector, which creates several equations based on the number of variables and constraints.
Solve these equations to find the values of x, y, z, and λ. From the solution set, identifying the highest point involves evaluating the z-coordinate since the question asks for the highest point in terms of altitude. Ensure that the solution satisfies both the original cone equation and the plane equation.