The profit function for a business is given by the equation [tex]P(x) = -4x^2 + 16x - 7[/tex], where [tex]x[/tex] is the number of items sold, in thousands, and [tex]P(x)[/tex] is the profit in thousands of dollars.
Calculate the maximum profit and the number of items that must be sold to achieve it.
Answer (3)
you can calculate the derivative of this function and find when it will be equal to zero since we know that such afunction as only one single tip for wich the slope is equal to zero. So finding for wich value of x the derivative of this function is zero, here, will give you the value of x for which you reach the maximum, so for how many item sold you'll reach it.
P'(x) = -8x + 16 P'(x) = 0 x = ? 0 = -8x +16 8x = 16 x = 2
P'(x) is the derivative of the finction you gave me and 2 is then the number of items sold in thousdans for wich you reach a maximum profit. Now to calculate that maximum profit, just substitute 2 for x in P(x).
P(2) = ? P = -4(2^2)+ 16(2) - 7 P = -4(4) + 32 - 7 P = -16 + 32 - 7 P = 9
So the maximum profit would be 9,000 $ for 2,000 items sold.
The maximum profit is $9,000, and 2,000 items must be sold to achieve it.
To find the maximum profit and the corresponding number of items sold, you need to determine the vertex of the quadratic function P ( x ) = − 4 x 2 + 16 x − 7 .
The profit function is in the form P ( x ) = a x 2 + b x + c , where a = − 4 , b = 16 , and c = − 7 .
The x-coordinate of the vertex is given by the formula x = − 2 a b . In this case:
x = − 2 ( − 4 ) 16 = − − 8 16 = 2
Now that you have the x-coordinate of the vertex (\( x = 2 \)) , you can find the corresponding y-coordinate (which represents the maximum profit). Plug x = 2 into the profit function:
P ( 2 ) = − 4 ( 2 ) 2 + 16 ( 2 ) − 7
P ( 2 ) = − 4 ( 4 ) + 32 − 7
P ( 2 ) = − 16 + 32 − 7
P ( 2 ) = 9
So, the maximum profit is $9,000 (since the profit is given in thousands of dollars), and this occurs when 2,000 items are sold.
The maximum profit is $9,000, and it occurs when 2,000 items are sold. This is determined by finding the vertex of the profit function. The number of items is represented in thousands within the function.
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