Find the number of units that must be produced and sold to yield the maximum profit, given the following equation for revenue and cost: R(x)=20x-0.5x^2, c(x)=6x+5
To find the number of units that must be produced and sold to yield the maximum profit, we need to derive the profit function from the revenue and cost functions.
The profit function (P) is given by subtracting the cost function from the revenue function:
P(x) = R(x) - c(x)
P(x) = (20x - 0.5x^2) - (6x + 5)
P(x) = 20x - 0.5x^2 - 6x - 5
P(x) = -0.5x^2 + 14x - 5
To find the number of units that yield the maximum profit, we need to find the x-value at which the derivative of the profit function equals zero. So, let's find the derivative of P(x).
dP/dx = -1x + 14
Now, set the derivative equal to zero and solve for x:
-1x + 14 = 0
x = 14
The number of units that must be produced and sold to yield the maximum profit is 14.