A boy is selling lemonade throughout the hot summer. Suppose the number of cups sold is given by the function, `n(X)=x2^(-x)+2` , where the price, *x*, in dollars determines the number of cups sold per day, *n*, in hundreds.

a) What is the price that maximizes the number of cups sold?

b) How many cups are sold at this price?

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In order to determine the maximum number of cups sold, we need to first find for what value of x (price) n(x) is maximized. To do so, first take the derivative of n(x):

`n'(x)=2^(-x)-x2^(-x)ln(2)`

Now set the derivative to 0 and solve for x:

`0=2^(-x)-x2^(-x)ln(2)`

`0=1-xln(2)`

`x=1/ln(2)=1.44`

a) Therefore, if they are sold at $1.44 the number sold is maximized.

b) `n(1.44)=1.44*2^(-1.44)+2=2.53~~3`

Approximately 3 cups are sold at a price of $1.44

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