The price ‘p’ per unit at which a company can sell all that it produces is given by p = 29 – x, where
‘x’ is the number of units produced. The total cost function C(x) = 45 + 11x. If P(x) = R(x) – C(x), is
the profit function then find the interval in which the profit is increasing and decreasing
To find the profit function, we need to find the revenue function. The revenue, R(x), is given by the product of the price per unit, p, and the number of units produced, x.
R(x) = p * x = (29 - x) * x = 29x - x^2
The profit function is then given by subtracting the total cost function, C(x), from the revenue function, R(x).
P(x) = R(x) - C(x) = (29x - x^2) - (45 + 11x) = 29x - x^2 - 45 - 11x = -x^2 + 18x - 45
To find the intervals where the profit is increasing or decreasing, we need to analyze the derivative of the profit function.
P'(x) = -2x + 18
To determine the intervals of increasing and decreasing profit, we need to find the critical points of the profit function by setting the derivative equal to zero and solving for x.
-2x + 18 = 0
2x = 18
x = 9
So the critical point is x = 9.
To determine whether the profit is increasing or decreasing on either side of x = 9, we can examine the sign of the derivative.
When x < 9:
P'(x) = -2x + 18 < 0
The derivative is negative, indicating that the profit is decreasing.
When x > 9:
P'(x) = -2x + 18 > 0
The derivative is positive, indicating that the profit is increasing.
Therefore, the profit is decreasing when x < 9 and increasing when x > 9.
To find the profit function, we need to find the revenue function. The revenue, R(x), is given by the product of the price per unit, p, and the number of units produced, x.
R(x) = p * x = (29 - x) * x = 29x - x^2
The profit function is then given by subtracting the total cost function, C(x), from the revenue function, R(x).
P(x) = R(x) - C(x) = (29x - x^2) - (45 + 11x) = 29x - x^2 - 45 - 11x = -x^2 + 18x - 45
To find the intervals where the profit is increasing or decreasing, we need to analyze the derivative of the profit function.
P'(x) = -2x + 18
To determine the intervals of increasing and decreasing profit, we need to find the critical points of the profit function by setting the derivative equal to zero and solving for x.
-2x + 18 = 0
2x = 18
x = 9
So the critical point is x = 9.
To determine whether the profit is increasing or decreasing on either side of x = 9, we can examine the sign of the derivative.
When x < 9:
P'(x) = -2x + 18 < 0
The derivative is negative, indicating that the profit is decreasing.
When x > 9:
P'(x) = -2x + 18 > 0
The derivative is positive, indicating that the profit is increasing.
Therefore, the profit is decreasing when x < 9 and increasing when x > 9.
The interval in which the profit is decreasing is when x < 9, and the interval in which the profit is increasing is when x > 9.
To find the interval in which the profit is increasing and decreasing, we need to determine the sign of the derivative of the profit function, P(x).
First, let's find the revenue function, R(x). The revenue is calculated by multiplying the price per unit (p) by the number of units sold (x). In this case, the price per unit is given by p = 29 - x. Therefore, the revenue function is:
R(x) = (29 - x) * x
Expanding this equation, we have:
R(x) = 29x - x^2
Now, let's substitute the revenue function and the cost function into the profit function:
P(x) = R(x) - C(x)
P(x) = (29x - x^2) - (45 + 11x)
Simplifying this equation, we get:
P(x) = 29x - x^2 - 45 - 11x
P(x) = -x^2 + 18x - 45
To find the intervals in which the profit is increasing or decreasing, we need to find the critical points. The critical points occur when the derivative of the profit function is equal to zero.
Taking the derivative of P(x) with respect to x, we get:
P'(x) = -2x + 18
Setting P'(x) equal to zero, we have:
-2x + 18 = 0
Solving for x, we find:
-2x = -18
x = 9
Now, we need to determine the intervals in which the profit is increasing and decreasing. We can do this by analyzing the sign of the derivative.
When x < 9, P'(x) = -2x + 18 is negative, indicating that the profit is decreasing.
When x > 9, P'(x) = -2x + 18 is positive, indicating that the profit is increasing.
Therefore, the interval in which the profit is decreasing is x < 9, and the interval in which the profit is increasing is x > 9.