Suppose the Sunglasses Hut Company
has a profit function given by p of q
where q is the number of thousands of pairs
of sunglasses sold and produced,
and p of q is the total profit,
in thousands of dollars,
from selling and producing q pairs of sunglasses.
We have three parts of this question.
Part a, find the marginal profit function,
which would be the first derivative
of our profit function p of q.
B, how many pairs of sunglasses, in thousands,
should be sold to maximize profits?
And we're asked to round to three decimal places.
And then finally c, what are the actual maximum profits,
in thousands, that can be expected?
So, we'll first find the marginal profit function,
which would be p prime of q,
which is equal to the derivative of
negative 0.03 q squared, so we'd multiply by two.
So, we'd have negative 0.06 times q
raised to the power of two minus one,
which would be q to the first, which is q.
Plus the derivative of three q, that'd be three.
So, plus three.
Minus the derivative of 36, which would be zero.
So this is our marginal profit function, p prime of q.
And now for part b, we're asked
how many pairs of sunglasses, in thousands,
should be sold to maximize profits.
So, the first step is going to be
to find the critical numbers,
which is where the first derivative
is equal to zero or undefined.
So, let's find our critical numbers on the next slide.
Again, we already know that p prime of q
is equal to negative 0.06q plus three.
So to find the critical numbers,
we'll set p prime of q equal to zero and solve for q.
If p of q does have a maximum value,
it will occur at a critical number.
Let's go ahead and add 0.06q to both sides.
That would give 0.06q equals three,
and then we'll divide both sides by 0.06.
Simplifying we have q equals
three divided by 0.06 is 50.
So, notice how we only have one critical number.
It might tempting to just assume
that this going to be where the profit is maximized,
but we really should apply it
to the first and second derivative test
to verify we have a maximum value at q equals 50.
So, for this example,
let's apply the second derivative test.
So we'll test the concavity at q equals 50,
to see if the function is concave up or concave down.
This will tell us whether we have a maximum value
at q equals 50.
But in reality, notice how the profit function,
p of q, is a quadratic function
with a negative leading coefficient.
So, we have a parabola that opens down.
So we are going to have a maximum value at q equals 50,
but let's go ahead and apply
the second derivative test anyway.
So, p double prime of q
would be equal to the derivative of the first derivative.
So the derivative of negative 0.06q
with respect to q would just be negative 0.06.
The derivative of three is zero.
So, notice how here this tells us
the second derivative is always negative,
so our function is always concave down.
If we want p double prime of 50,
would still be equal to negative 0.06,
which means our function is concave down
at the critical number 50.
So, if the function is concave down,
it would look something like this
and therefore at this critical number,
q equals 50, we do have a high point,
or a maximum function value,
which means 50,
not just 50, but 50,000 pairs of sunglasses
should be sold to maximize profits.
Now for part c, we're asked
what are the actual maximum profits that can be expected.
So, to find the actual maximum profits,
we want to evaluate the profit function, p of q,
at q equals 50.
So, p of 50 is going to be equal to
negative 0.03
times 50 squared
plus three times 50
minus 36.
Now we go to the calculator.
Negative 0.03 times 50 squared
plus three times 50
minus 36
is equal to 39.
So, p of 50 equals 39,
but remember, p of q is in thousands of dollars.
So the maximum profit
is not $39, but $39,000.
Before we go, let's check our answers graphically
by analyzing our profit function p of q.
Notice how, as I mentioned before,
we do have a quadratic function.
Our leading coefficient is negative
so the parabola opens down as we see here.
Here's the high point on the graph.
Notice how this occurs when q is equal to 50,
verifying to maximize profit,
they want to sell 50,000 pairs of sunglasses.
Notice how when q is 50,
p of q, the total profit,
is equal to $39,000.
I hope you found this helpful.