IXL - Introduction to optimization (Calculus practice)

Maximize the quantity

–

subject to the constraints

–

and

Complete the sentences with exact, simplified answers.

The maximum value is

It occurs when

and

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Learn with an example

question

Maximize the quantity

subject to the constraint

200

Complete the sentences with exact, simplified answers.

The maximum value is

It occurs when

and

key idea

If a function has a maximum value, then the maximum occurs either at a critical point or at an endpoint of the function's domain.

solution

You want to maximize the quantity

To do this, first express the quantity as a function of a single variable. You can use the constraint

200

to solve for

200

–

100

–

Now, replace

in the expression

to write it in terms of

only.

(

100

–

)

100

–

So, if

200

then maximizing the quantity

is equivalent to maximizing the function

(

)

100

–

To maximize

(

)

first find its critical points.

(

)

100

–

′

(

)

100

–

100

–

Since

′

(

)

is defined for all

(

)

only has a critical point at

Note that

′′

(

)

–

is negative for all

so the graph of

must be concave down and the critical point at

is at an absolute maximum. That is, the maximum value of

(

)

occurs when

This means the maximum value of

must also occur when

To find the corresponding

-value,

use the constraint

200

(

)

200

100

200

100

Therefore, the quantity

is maximized when

and

To find the maximum value, you can plug these values for

and

into the expression

or you can evaluate

(

)

(

)

(

)

1,250

So, the maximum value is 1,250, and it occurs when

and

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