(O4) Constrained Optimization#

By the end of the lesson you will be able to:

  • use derivatives to solve constrained optimization problems.

Example 1#

Find the maximum value of \(Q=xy\) provided \(x+y=2\).

Step 1: Determine the Objective and Constraint Equations

  • Objective Equation - what we want to maximize or minimize.

  • Constraint Equation - the equation that puts some restriction on the variables.

Step 2: Solve the Contraint Equation for one of the variables (whichever is easier).

Step 3: Plug this into Objective Equation, giving us a one-variable function to optimize.

Step 4: Optimize the resulting function. (Find and classify the critical numbers.)

Step 5: Answer the question.

Example 2#

Find the values of \(x\) and \(y\) that maximize \(V=xy^2\) provided \(2y^2+4xy=60\).

Step 1: Determine the Objective and Constraint Equations

  • Objective Equation - what we want to maximize or minimize.

  • Constraint Equation - the equation that puts some restriction on the variables.

Step 2: Solve the Contraint Equation for one of the variables (whichever is easier).

Step 3: Plug this into Objective Equation, giving us a one-variable function to optimize.

Step 4: Optimize the resulting function. (Find and classify the critical numbers.)

Step 5: Answer the question.