What is the price that maximze the number of cups sold and how many cups are sold at this price?
A girl is selling coffee throughout winter. Suppose the number of cups sold is given by the function n(x)= x^(2-x) + 2. where the price, x in dollars determines the number of cups sold per day, n, in hundreds.
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You should notice that this is an optimization problem that involves derivative of the given function.
You should come up with the substitution `y=n(x)` such that:
`y = x^(2-x) + 2`
Since the function `x^(2-x)` is neither an exponential function of form `a^x` , nor a power function of form `x^a,` you need to use the next formula of differentiation:
`x^(2-x) = (2-x)*x^(2-x-1) + x^(2-x)*ln x`
Notice that the function is considered first a power function and it is consequently differentiated then the function is differentiated like it would be an exponential function.
Hence, differentiating the function with respect to x yields
`(dy)/(dx) = (2-x)*x^(2-x-1) + x^(2-x)*ln x`
`(dy)/(dx) = (2-x)*x^(1-x) + x^(2-x)*ln x`
You need to solve the equation `(dy)/(dx) = 0` to find x value that maximizes the function such that:
`(2-x)*x^(1-x) + x^(2-x)*ln x = 0`
You need to factor out `x^(1-x) ` such that:
`x^(1-x)*(2 - x + x*ln x) = 0`
`x^(1-x) = 0 =gt x = 0`
`2 - x + x*ln x = 0 =gt x - 2 = x*ln x`
Notice that the graph of function `x*ln x ` (orange curve) does not intercept the graph of x-2, the black line, hence the equation `x - 2 = x*ln x` has no solution.
If the price x is 0 dollars, there is no sell done, hence the model of function selected to represent the number of cups sold is inconsistent.
n(x)= x^(2-x) + 2
==> n(x)= x^2 / x^x + 2
We need to find the maximum value.
`==gt n'(x)= ((x^x)(x^2)'- (x^2)(x^x)')/ x^(2x) `
`=gt n'(x)= ((x^x)(x^2 ln 2) - (x^2)(x^x)lnx)/(x^(2x)) `
`==gt n'(x)=((x^x)(x^2 )( ln 2 - lnx))/ (x^(2x))`
`` Now we will find the derivative's zero.
==> x= 0
==> ln2 - lnx = 0 ==> x = 2
We will not consider x= 0 because the price can not be 0 in order to profit.
Then, the price that maximizes the selling is x= 2
Now we will find the number of cups sold.
==> n(2)= 2^(2-2) + 2 = 1 + 2= 3
==> Then the total number of cups is 300 cups.
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