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with all math functions spelled out and without any math notation or math font:
We're given the cost function C of x and the demand function P of x. We want to find the
production level that will maximize profit. Since profit is equal to revenue minus cost,
let's begin by determining the revenue function. The revenue
function R of x is equal to the price per unit, which is P of x,
multiplied by the quantity of items produced and sold, which is x. But we know P of x
equals one thousand five hundred, so the revenue function is one thousand five hundred times x.
We are also given the cost function C of x, which means our profit function P of x would be equal
to R of x, or one thousand five hundred times x, minus C of x, the cost function. So we would have
minus the quantity two thousand six hundred plus five hundred x plus zero point nine x squared.
Now we’ll simplify the parentheses and combine like terms. If it’s helpful
because of the subtraction here, we can think of distributing a negative
one. So we would have P of x equal to one thousand five hundred times x
minus two thousand six hundred minus five hundred x minus zero point nine x squared.
Combining like terms, our profit function P of x is equal to negative zero point nine x
squared. Combining the x terms, we’d have plus one thousand x minus two thousand six hundred.
Now that we have the profit function, we can determine the production level or the
value of x that will maximize the profit. The first step is to find the critical
numbers of the profit function, and we can do this in one of two ways. One way
is to determine where the first derivative is equal to zero or undefined. The second
method would be to determine when the marginal revenue and the marginal cost
are equal. Let's determine where the first derivative is equal to zero or undefined.
The derivative P prime of x is equal to the derivative of negative zero point nine x squared.
So we multiply by two, which gives us negative one point eight times x, plus the derivative
of one thousand x, which is one thousand, minus the derivative of two thousand six hundred,
which is zero. Notice how the derivative function is linear, which is never undefined.
To find the critical number, we'll set this equal to zero and solve for
x. So subtracting one thousand from both sides, we have negative one point eight x
equals negative one thousand. Now we’ll divide both sides by
negative one point eight, which gives us approximately five hundred fifty-six.
So we only have one critical number, but we should still verify this value of x will maximize the
profit and not minimize the profit. We recognize that the profit function is a quadratic where the
coefficient is negative, so we have a parabola that opens down. Therefore, the profit function
is always concave down. This verifies that at this x value, we would have a maximum.
Another option would be to use either the first or second derivative test to determine whether
we have a maximum or minimum at this x value. Notice that if we found the second derivative
function P double prime of x, it would be equal to just negative one point eight, which
again means the function is always concave down. This indicates that
at this critical number, we would have a high point or a maximum function value.
However, because x represents the quantity produced and sold, we should round this to
the nearest whole number. So we’ll round this to five hundred fifty-six. Therefore, we can say the
profit will be maximized at a production level of five hundred fifty-six units.
I hope you found this helpful!