Q. 2 - Calculus

Question

Extrema on a closed and bounded set: How could we find the maximum and minimum values of the function $f(x, y) = 3x - 4y$if we restrict the domain to the disk $x^2+y^2\le 1$?

Step-by-Step Solution

Answer

We can find the maximum and minimum values of the function $f(x, y) = 3x - 4y$if we restrict the domain to the disk $x^2+y^2\le 1$ by using the Extreme Value Theorem for a function of two Variables.

Step 1 . Given information

We need to explain that how can we find the maximum and minimum values of the function $f(x, y) = 3x - 4y$ if we restrict the domain to the disk $x^2+y^2\le 1$.

Step 2 . Method used in finding the maximum and the minimum values of the function f ( x ,   y )   =   3 x   −   4 y if we restrict the domain to the disk.

We can use the Extreme Value Theorem for a function of two variables to find

the maximum and minimum values of the function $f(x, y) = 3x - 4y$ if we restrict the domain to the disk.

Step 3 . Steps for using the Extreme Value Theorem for a function of two variables:

Given a continuous function $f\left(x,y\right)=3x-4y$ defined on a closed and bounded set $S$,we may use the following outline to find those points $(x_M, y_M)$and $(x_m, y_m)$which maximize and minimize $f\left(x,y\right)=3x-4y$ on $S$:

• Find the stationary points and other critical points of $f$ .
• Select only those critical points that lie in $S$.
• Evaluate the function at each of the critical points found in step $2$.
• Use the method of Lagrange multipliers to locate the points on the boundary of $S$ that maximize and minimize $f$ .
• Evaluate the function at each of the critical points on the boundary of $S$.
• Use the extrema from steps $3$ and $5$ to find the maximum and minimum values of the function $f$ on $S$.