Find the maximum profit that a company can make, if the profit function is given by p(x) = 41 - 72x - 18x².
The maximum profit that the company can make is 113.
Explanation
This question involves finding the maximum value of a profit function using differentiation. Following the method shown in the textbook context, we need to find the critical point by setting the first derivative to zero, then verify it's a maximum using the second derivative test. The context demonstrates this approach in the example where P'(x) = 0 gives the critical point and P''(x) < 0 confirms it's a point of maxima.
Solution Steps
Step 1: Given profit function is p(x) = 41 - 72x - 18x²
Step 2: Find the first derivative: p'(x) = -72 - 36x
Step 3: Set p'(x) = 0 to find critical point: -72 - 36x = 0, which gives x = -2
Step 4: Find the second derivative: p''(x) = -36
Step 5: Since p''(x) = -36 < 0, the function has maximum at x = -2
Step 6: Calculate maximum profit: p(-2) = 41 - 72(-2) - 18(-2)² = 41 + 144 - 72 = 113