# 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...

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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