Find the value of x that maximizes y = 20x – x Find the value of x that minimizes y = x Given TC = x Given TR…

Question

Find the value of x that maximizes y = 20x – x Find the value of x that minimizes y = x Given TC = x Given TR = 100x – x A firm's profit function is π = 120x – x

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Here are the solutions to the practice problems.

41. Find the value of $x$ that maximizes $y = 20x - x^2$ .

To find the maximum value, we take the first derivative of the function $y$ with respect to $x$ .
$\frac{dy}{dx} = \frac{d}{dx}(20x - x^2) = 20 - 2x$
Set the first derivative equal to zero and solve for $x$ .
$20 - 2x = 0$ $2x = 20$ $x = 10$
The value of $x$ that maximizes $y$ is $\boxed{\text{10}}$ .

42. Find the value of $x$ that minimizes $y = x^2 + 4x + 7$ .
To find the minimum value, we take the first derivative of the function $y$ with respect to $x$ .
$\frac{dy}{dx} = \frac{d}{dx}(x^2 + 4x + 7) = 2x + 4$
Set the first derivative equal to zero and solve for $x$ .
$2x + 4 = 0$ $2x = -4$ $x = -2$
The value of $x$ that minimizes $y$ is $\boxed{\text{-2}}$ .

43. Given $TC = x^2 + 10x + 50$ , find the Marginal Cost.
Marginal Cost (MC) is the derivative of the Total Cost (TC) function with respect to $x$ .
$MC = \frac{d(TC)}{dx}$
Differentiate the given TC function.
$MC = \frac{d}{dx}(x^2 + 10x + 50) = 2x + 10$
The Marginal Cost is $\boxed{2x + 10}$ .

44. Given $TR = 100x - x^2$ , find the Marginal Revenue.
Marginal Revenue (MR) is the derivative of the Total Revenue (TR) function with respect to $x$ .
$MR = \frac{d(TR)}{dx}$
Differentiate the given TR function.
$MR = \frac{d}{dx}(100x - x^2) = 100 - 2x$
The Marginal Revenue is $\boxed{100 - 2x}$ .

45. A firm's profit function is $\pi = 120x - x^2 - 200$ . Determine the output level that maximizes profit.
To maximize profit, we take the first derivative of the profit function $\pi$ with respect to $x$ .
$\frac{d\pi}{dx} = \frac{d}{dx}(120x - x^2 - 200) = 120 - 2x$
Set the first derivative equal to zero and solve for $x$ .
$120 - 2x = 0$ $2x = 120$ $x = 60$
The output level that maximizes profit is $\boxed{\text{60}}$ .

Find the value of x that maximizes y = 20x – x Find the value of x that minimizes y = x Given TC = x Given TR = 100x – x A firm's profit function is π = 120x – x

Step-by-step solution

To find the maximum value, we take the first derivative of the function $y$ with respect to $x$ .

Set the first derivative equal to zero and solve for $x$ .

To find the minimum value, we take the first derivative of the function $y$ with respect to $x$ .

Set the first derivative equal to zero and solve for $x$ .

Marginal Cost (MC) is the derivative of the Total Cost (TC) function with respect to $x$ .

Differentiate the given TC function.

Marginal Revenue (MR) is the derivative of the Total Revenue (TR) function with respect to $x$ .

Differentiate the given TR function.

To maximize profit, we take the first derivative of the profit function $\pi$ with respect to $x$ .

Set the first derivative equal to zero and solve for $x$ .

Still stuck on this one?

More Mathematics Questions

Find the value of x that maximizes y = 20x – x Find the value of x that minimizes y = x Given TC = x Given TR = 100x – x A firm's profit function is π = 120x – x

Step-by-step solution

To find the maximum value, we take the first derivative of the function yyy with respect to xxx.

Set the first derivative equal to zero and solve for xxx.

To find the minimum value, we take the first derivative of the function yyy with respect to xxx.

Set the first derivative equal to zero and solve for xxx.

Marginal Cost (MC) is the derivative of the Total Cost (TC) function with respect to xxx.

Differentiate the given TC function.

Marginal Revenue (MR) is the derivative of the Total Revenue (TR) function with respect to xxx.

Differentiate the given TR function.

To maximize profit, we take the first derivative of the profit function π\piπ with respect to xxx.

Set the first derivative equal to zero and solve for xxx.

Still stuck on this one?

More Mathematics Questions

To find the maximum value, we take the first derivative of the function $y$ with respect to $x$ .

Set the first derivative equal to zero and solve for $x$ .

To find the minimum value, we take the first derivative of the function $y$ with respect to $x$ .

Set the first derivative equal to zero and solve for $x$ .

Marginal Cost (MC) is the derivative of the Total Cost (TC) function with respect to $x$ .

Marginal Revenue (MR) is the derivative of the Total Revenue (TR) function with respect to $x$ .

To maximize profit, we take the first derivative of the profit function $\pi$ with respect to $x$ .

Set the first derivative equal to zero and solve for $x$ .