We're given the cost function
C of X equals 62,500
plus 500X plus X squared
where C of X is the total cost in dollars
and X is the production level.
We want to answer the following questions.
A, what is the cost at the production level of 1,050?
B, what is the average cost
at the production level of 1,050?
C, What is the marginal cost
at the production level of 1,050?
And then D, what is the production level
that will minimize the average cost?
And then E, what is the minimum average cost?
So to begin, to determine the cost or total cost
at a production level of 1,050,
we want to evaluate the cost function at 1,050.
So this will give us C of 1,050
equals 62,500
plus 500 times 1,050
plus 1,050 squared.
Let's go and evaluate this on the calculator.
So the total cost would be $1,690,000.
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Part B, what is the average cost
at the production level of 1,050?
Well, the average cost function
is equal to the total cost function
divided by the production level,
or C of X divided by X.
Let's first find the average cost function.
When dividing by a monomial or a single term,
we divide each term in the numerator by the denominator,
which would give us 62,500
divided by X plus 500X divided by X
plus X squared divided by X.
So the average cost function
is equal to 62,500 divided by X
plus 500 plus X
and now we'll evaluate this function at X equals 1,050
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and now we'll go back to the calculator.
So the average cost to the nearest cent
would be approximately $1,609.52.
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Part C, what is the marginal cost
at the production level of 1,050?
The marginal cost function
is the derivative of the cost function
and it's used to approximate the cost
of the X plus oneth item, so in this case,
it would give us the cost of the next item
when the production level is 1,050.
So first, the marginal cost function, C prime of X
with equal to the derivative of 62,500 would be zero
plus the derivative of 500X, that would be 500,
plus the derivative of X squared, that would be 2X.
Now we'll evaluate this at X equals 1,050.
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This would be 500 plus 2,100, which would give us 2,600.
So the marginal cost at X equals
1,050 is $2,600
which, again, is the cost to produce the next item
when the production level is 1,050.
Part D, what is the production level
that will minimize the average cost?
So going back to part B,
notice how this is the average cost function,
so to minimize this function,
we first want to find the critical numbers.
We can do this two ways.
We can either determine where the derivative
of the average cost function is equal to zero or undefined,
or when the average cost and marginal cost are equal.
Let's go ahead and use the calculus method here,
so for the next step we'll find the derivative
of this average cost function,
but before we do this though, let's write this first term
where this is X of the first in the denominator,
so we can move this up.
We can write this as 62,500 X to the power of negative one.
So the derivative of this
would be equal to negative 62,500
X to the power of negative two
plus zero and then plus one
which is the same as negative 62,500
divided by X squared plus one
and now we'll set this equal to zero and solve.
Notice how this function would be undefined
when X equals zero, but when X equals zero
the production level would be zero.
So to solve this equation here,
let's multiply through by X squared,
so that would give us a negative 62,500
plus X squared equals zero.
Solving for X squared, we have X squared equals 62,500
and now we'll square root both sides of the equation.
Again, because X is the production level,
we know X can't be negative,
so we're only concerned about the positive
or principal square root here.
So going back to the calculator,
the square root of 62,500
is 250.
So we only have one feasible critical number
but to verify that at this X value,
this will minimize the average cost function,
we needed to use the first derivative
or second derivative test.
So let's go ahead and use the second derivative test
to determine whether the function is concave up
or concave down at this value of X.
So here's the first derivative function,
so if we find the second derivative function,
notice how we'd multiply by negative two,
which would give us positive 125,000
times X to the power of negative three
and notice this second derivative function at 250
would be positive, or greater than zero,
which means the function is concave up
which is good news because that means
at X equals 250, we do have a minimum function value.
So the production value of 250
will minimize the average cost
and finally for the last question, or part E,
we want to determine the minimum average cost.
Well again, the average cost function
was equal to 62,500 divided by X
plus 500 plus X
and we just found that this function is minimized
when X equals 250, so to find the minimum average cost,
we'll evaluate this function at X equals 250.
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So notice here we have plus 750
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so the minimum average cost per item is $1,000.
I hope you found this helpful.