Las ganancias de un negocio se pueden determinar al restar los costos de los ingresos. Supongamos que la ganancia de un negocio son representadas por la función: R(x): 5x-0.01x^2 y los costoa de fabricaa del producto son representados por C(x): 100+2x donde x es el numero de unidades del producto. Determine la función ganancias. Halle una función P(x) que represente las ganancias de la empresa y determine la ganancia máxima.

Función de beneficios = P(x) =-0.01x² + 3x - 100

Ganancias máximos = 125

Y ocurre cuando x = 150 unidades

Profits Function = P(x) = -0.01x² + 3x - 100

Maximum profits = 125

And it occurs when x = 150 units

Step-by-step explanation:

Ganancias = Ingresos - Costos

Ingresos = R (x) = 5x - 0.01x²

Costos = C (x) = 100 + 2x

Función de ganancias = P (x) = R (x) - C (x)

= (5x - 0.01x²) - (100 + 2x)

= 5x - 0.01x² - 100 - 2x

= 3x - 0.01x² - 100

P (x) = -0.01x² + 3x - 100

Para obtener las máximas ganancias, utilizamos el análisis de diferenciación para la función de ganancias.

En el nivel de Ganancias máximo, (dP / dx) = 0 y (d²P / dx²) < 0

P(x) = -0.01x² + 3x - 100

(dP/dx) = -0.02x + 3

Al máximo Ganancias, (dP/dx) = 0

(dP/dx) = -0.02x + 3 = 0

x = (3/0.02) = 150 unidades

Este es el nivel de producto que corresponde a las ganancias máximas.

Para verificar si este es realmente el punto máximo de la función de ganancias,

(dP/dx) = -0.02x + 3

(d²P/dx²) = -0.02 < 0 (lo que demuestra que realmente es el punto de máximo ganancias).

Por lo tanto, la ganancia máxima ocurre cuando x = 150 unidades

P(x) = -0.01x² + 3x - 100

P(x) = -0.01(150²) + 3(150) - 100 = 125

¡¡¡Espero que esto ayude!!!

English Translation

The profits of a business can be determined by subtracting the costs from the income. Suppose that the revenue of a business is represented by the function:

R(x) = 5x - 0.01x²

and the manufacturing costs of the product are represented by

C(x) = 100 + 2x

where x is the number of units of the product. Determine the profits function. Find a function P (x) that represents the profits of the firm and determine the maximum profit.

Solution

Profits = Revenue - Costs

Revenue = R(x) = 5x - 0.01x²

Costs = C(x) = 100 + 2x

Profits function = P(x) = R(x) - C(x)

= (5x - 0.01x²) - (100 + 2x)

= 5x - 0.01x² - 100 - 2x

= 3x - 0.01x² - 100

P(x) = -0.01x² + 3x - 100

To obtain maximum profits, we use differentiation analysis for the profit function.

At the maximum profit level, (dP/dx) = 0 and (d²P/dx²) < 0

P(x) = -0.01x² + 3x - 100

(dP/dx) = -0.02x + 3

At maximum profit, (dP/dx) = 0

(dP/dx) = -0.02x + 3 = 0

x = (3/0.02) = 150 units

This is the product level that corresponds to maximum profits.

To check if this is truly the maximum point of the profit function,

(dP/dx) = -0.02x + 3

(d²P/dx²) = -0.02 < 0 (which shows that it truly is the maximum profit point.

Hence, maximum profit occurs when x = 150 units

P(x) = -0.01x² + 3x - 100

P(x) = -0.01(150²) + 3(150) - 100 = 125

Hope this Helps!!!

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