How many units should this company sell to maximize their revenue if the revenue function was -(x^2/100)+50x using derivatives?
the derivative is
-x/50 + 50
maximum revenue occurs when that is zero.
To find the number of units a company should sell to maximize their revenue, you can use derivatives. The revenue function is given as -(x^2/100) + 50x.
Step 1: Write the revenue function equation: R(x) = -(x^2/100) + 50x.
Step 2: Take the derivative of the revenue function with respect to x. Differentiating the equation will give us the slope of the function at any given point, which is helpful in identifying the maximum or minimum point.
To differentiate the equation, we treat each term separately:
dR(x)/dx = d/dx (-(x^2/100)) + d/dx (50x).
Differentiating the equation, we get:
dR(x)/dx = [(-2/100) * x^(2-1)] + [50 * 1]
= (-2/100) * x + 50
= -x/50 + 50.
Step 3: Set the derivative equal to zero and solve for x to find the critical points. At these points, the slope of the function is zero and could correspond to a maximum or minimum point.
-x/50 + 50 = 0.
Simplifying the equation, we get:
-x/50 = -50.
Multiply both sides by 50:
-x = -50 * 50.
Calculating, we have:
x = 2500.
So, the critical point is x = 2500.
Step 4: Check the second derivative to identify whether the critical point is a maximum or minimum.
To do this, we take the second derivative of the revenue function:
d²R(x)/dx² = d/dx (-x/50 + 50).
Differentiating the equation, we get:
d²R(x)/dx² = -1/50.
Since the second derivative is negative, this means that the critical point corresponds to a maximum.
Therefore, the company should sell 2500 units to maximize their revenue.