Question

2. The profit when selling 'X' pairs of shoes is defined by the function P(x)=-8x2 +80x-192. How many pairs of shoes should be sold to maximize the profit?

Answer 1

The maximum profit is achieved when 5 pairs of shoes are sold.

Explanation:

To find the maximum profit, we need to find the value of x that maximizes the function P(x). One way to do this is to use calculus.

First, we take the derivative of P(x) with respect to x:

P'(x) = -16x + 80

Then, we set P'(x) equal to zero to find the critical points:

-16x + 80 = 0

Solving for x, we get:

x = 5

Now we need to check if this critical point is a maximum or a minimum. To do this, we take the second derivative of P(x) with respect to x:

P''(x) = -16

Since P''(x) is negative for all values of x, we know that the critical point x = 5 corresponds to a maximum. Therefore, the maximum profit is achieved when 5 pairs of shoes are sold.