Question

The monthly profit, P (x), of a sportswear company, in thousands of dollars, is represented by the quadratic function P (x) = -2x^2 + 12x - 4, where x is the amount spent on advertising, in thousands of dollars. a) determine the company’s maximum monthly profit b) determine the amount spent on advertising to achieve the maximum profit [can someone help me on this? you don’t have to give me the answer u can just lmk how to do it]

Answer 1

• the amount spent (in thousands) is the x-coordinate of the vertex
• the maximum profit (in thousands) is the y-coordinate of the vertex

Explanation:

The function describes a downward-opening parabola. (It is 2nd-degree with a negative leading coefficient.) So, the vertex of the graph will answer both questions at once.

You can put the function into vertex form by factoring the leading coefficient (-2) from the x-terms, adding the square of half the x-coefficient inside parentheses (and subtracting the same quantity outside parentheses), then writing the result in the form ...

P(x) = -2(x -h)^2 + k

The vertex is (h, k), so the answer is "h" thousand dollars spent on advertising will result in a maximum monthly profit of "k" thousand dollars.

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The first two steps look like ...

P(x) = -2(x^2 -6x) -4

P(x) = -2(x^2 -6x +9) -4 -(-2)(9)

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It can also be useful to use a graphing calculator to make a graph of the function.