2. Economics A cost function relates the total cost \( (C) \) of production to the number of items \( (x) \) produced per day. Acme Industries has a cost function \( C_{1}=300 x+ \) 500 , while Amalgamated Widgets' cost function is \( C_{2}=420 x+250 \). Using these facts, find a counterexample to this conjecture. For any value of \( x \), Acme Industries has lower costs than Amalgamated Widgets.

To determine whether the conjecture "For any value of \( x \), Acme Industries has lower costs than Amalgamated Widgets" is true, we need to compare the cost functions of both companies for various values of \( x \). The cost functions are given as follows: - Acme Industries: \( C_1 = 300x + 500 \) - Amalgamated Widgets: \( C_2 = 420x + 250 \) To find a counterexample, we can set up the inequality: \[ C_1 < C_2 \] Substituting the cost functions into the inequality gives: \[ 300x + 500 < 420x + 250 \] Now, we can simplify this inequality: 1. Subtract \( 300x \) from both sides: \[ 500 < 120x + 250 \] 2. Subtract \( 250 \) from both sides: \[ 250 < 120x \] 3. Divide both sides by \( 120 \): \[ \frac{250}{120} < x \] 4. Simplifying \( \frac{250}{120} \): \[ \frac{250}{120} = \frac{25}{12} \approx 2.0833 \] This means that for any \( x \) greater than \( \frac{25}{12} \) (approximately 2.0833), Acme Industries does not have lower costs than Amalgamated Widgets. ### Counterexample Let's choose \( x = 3 \) (which is greater than \( \frac{25}{12} \)): - Calculate \( C_1 \) for \( x = 3 \): \[ C_1 = 300(3) + 500 = 900 + 500 = 1400 \] - Calculate \( C_2 \) for \( x = 3 \): \[ C_2 = 420(3) + 250 = 1260 + 250 = 1510 \] In this case, we find: - \( C_1 = 1400 \) - \( C_2 = 1510 \) Thus, for \( x = 3 \), Acme Industries has lower costs than Amalgamated Widgets. Now, let's try \( x = 4 \): - Calculate \( C_1 \) for \( x = 4 \): \[ C_1 = 300(4) + 500 = 1200 + 500 = 1700 \] - Calculate \( C_2 \) for \( x = 4 \): \[ C_2 = 420(4) + 250 = 1680 + 250 = 1930 \] Again, we find: - \( C_1 = 1700 \) - \( C_2 = 1930 \) Now, let's try \( x = 5 \): - Calculate \( C_1 \) for \( x = 5 \): \[ C_1 = 300(5) + 500 = 1500 + 500 = 2000 \] - Calculate \( C_2 \) for \( x = 5 \): \[ C_2 = 420(5) + 250 = 2100 + 250 = 2350 \] Again, we find: - \( C_1 = 2000 \) - \( C_2 = 2350 \) ### Conclusion To find a counterexample, we need to find a value of \( x \) such that \( C_1 \geq C_2 \). Let's try \( x = 2 \): - Calculate \( C_1 \) for \( x = 2 \): \[ C_1 = 300(2) + 500 = 600 + 500 = 1100 \] - Calculate \( C_2 \) for \( x = 2 \): \[ C_2 = 420(2) + 250 = 840 + 250 = 1090 \] In this case, we find: - \( C_1 = 1100 \) - \( C_2 = 1090 \) Thus, for \( x = 2 \), Acme Industries has higher costs than Amalgamated Widgets, which serves as a counterexample to the conjecture. Therefore, the conjecture is false.