Maximizing Profit: Calculating Optimal Production
To maximize profit, a firm should produce at the level where Marginal Cost (MC) equals Marginal Revenue (MR). The profit is maximized when this condition holds, provided the price is greater than the average variable cost (which is implied here as we seek maximization).
1. Define Cost and Revenue Functions
- Cost Function: $ C(q) = 5q^2 $
- Revenue Function: $ R(q) = \text{Price} \times q = 50q $
2. Calculate Marginal Cost (MC)
Marginal Cost is the derivative of the Cost Function with respect to quantity ($q$): $ MC = \frac{dC}{dq} = \frac{d}{dq}(5q^2) = 10q $
3. Calculate Marginal Revenue (MR)
Marginal Revenue is the derivative of the Revenue Function with respect to quantity ($q$): $ MR = \frac{dR}{dq} = \frac{d}{dq}(50q) = 50 $
4. Find Production Quantity for Maximum Profit
Set MC equal to MR to find the profit-maximizing quantity ($q$): $ MC = MR $ $ 10q = 50 $ $ q = \frac{50}{10} $ $ q = 5 $
The second-order condition for profit maximization requires the second derivative of the profit function to be negative. Profit $ \Pi(q) = R(q) - C(q) = 50q - 5q^2 $. $ \frac{d\Pi}{dq} = 50 - 10q $ $ \frac{d^2\Pi}{dq^2} = -10 $ Since $ -10 < 0 $, the profit is indeed maximized at $ q=5 $.
Therefore, the firm should produce 5 units to maximize profit.
Important Questions from Maxima & Minima
The maximum value attained by the function \(f\left( x \right) = x\left( {x - 1} \right)\left( {x - 2} \right)\) in the interval [1, 2] is _____.
The function ƒ(x) = 3x (x - 2) has:
- The minimum value of the function $f(x) = |x| + |2x + 3|$ for real $x$ is ________ (rounded off to 1 decimal place).
- The value of the determinant of the Hessian of $f(x,y) = x^2 + y^2 + xy - 8x - 7y$ at its stationary point is
- A rectangular box with an open top has an outer surface area of 12 m$^2$. The length of the base is twice the width. If the volume of the box is to be maximised, then the ratio of length to height should be ____________. (answer in integer)
