Optimization Problems Involving Numbers

Solved Problems

Solution.

The objective function is written in the form

where $x$ and $y$ are the two numbers.

As $x+y=12$ we can write

Compute the derivative:

Determine the critical points:

At $$ and $$ the objective function is equal to zero.

When $$ the value of $$ is

At this point, the objective function attains the maximum value:

Solution.

Let $$ and $$ be the two numbers. The constraint equation is written in the form

The objective function is given by

Find the derivative and determine the critical points:

Thus, the function has two critical points $$ and $$ We should take only positive number $$

Using the First Derivative Test, one can show that $$ is a point of minimum.

The second number is $$

Solution.

Let $$ and $$ be the two numbers. The objective function is given by

As $$ we substitute $$ in the function above.

Differentiate $$

It is clear that the positive critical value is only $$ Using the First Derivative Test, one can show that $$ is a point of local maximum.

Respectively, the other number is $$

Solution.

Let the two numbers be $$ and $$ The objective function is written as

The constraint equation has the form

Hence

Expanding $$ we obtain:

Differentiate:

Find the critical points:

When $$ then $$ so the objective function is equal to zero in this case.

Note that the second derivative is

Hence, the second derivative is negative for $$ that is the point $$ is a point of maximum of the objective function.

The other number $$ is equal to

Solution.

We write the objective function in the form

where $$ are two positive numbers.

As $$ we can plug in $$ into the objective function.

Differentiate $$

Determine the critical points:

There are total 3 critical points: $$ We calculate the values of the objective function at these points:

Thus, the maximum value $$ is attained at $$ $$