Answer :
To find the maximum profit for the given profit function, we have the equation [tex]\( P(x) = -4x^2 + 16x - 7 \)[/tex]. This equation is a quadratic in the standard form [tex]\( ax^2 + bx + c \)[/tex].
Since this is a quadratic function, its graph is a parabola. Because the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\( -4 \)[/tex]), the parabola opens downward, which means it has a maximum point at the vertex.
To find the x-coordinate of the vertex for a parabola described by the equation [tex]\( ax^2 + bx + c \)[/tex], we use the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Let's apply this formula:
- [tex]\( a = -4 \)[/tex]
- [tex]\( b = 16 \)[/tex]
Plug these values into the vertex formula:
[tex]\[ x = -\frac{16}{2 \times -4} \][/tex]
[tex]\[ x = -\frac{16}{-8} \][/tex]
[tex]\[ x = 2 \][/tex]
The x-coordinate of the vertex is [tex]\( x = 2 \)[/tex], which corresponds to 2,000 items since [tex]\( x \)[/tex] is in thousands.
Next, we calculate the maximum profit by substituting this x-coordinate back into the profit function [tex]\( P(x) \)[/tex]:
[tex]\[ P(2) = -4(2)^2 + 16(2) - 7 \][/tex]
[tex]\[ P(2) = -4 \times 4 + 16 \times 2 - 7 \][/tex]
[tex]\[ P(2) = -16 + 32 - 7 \][/tex]
[tex]\[ P(2) = 16 - 7 \][/tex]
[tex]\[ P(2) = 9 \][/tex]
Hence, the maximum profit is [tex]\( 9 \)[/tex] thousand dollars, which means [tex]$9,000.
Therefore, the maximum profit is $[/tex]\[tex]$9,000$[/tex].
Since this is a quadratic function, its graph is a parabola. Because the coefficient of [tex]\( x^2 \)[/tex] is negative ([tex]\( -4 \)[/tex]), the parabola opens downward, which means it has a maximum point at the vertex.
To find the x-coordinate of the vertex for a parabola described by the equation [tex]\( ax^2 + bx + c \)[/tex], we use the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Let's apply this formula:
- [tex]\( a = -4 \)[/tex]
- [tex]\( b = 16 \)[/tex]
Plug these values into the vertex formula:
[tex]\[ x = -\frac{16}{2 \times -4} \][/tex]
[tex]\[ x = -\frac{16}{-8} \][/tex]
[tex]\[ x = 2 \][/tex]
The x-coordinate of the vertex is [tex]\( x = 2 \)[/tex], which corresponds to 2,000 items since [tex]\( x \)[/tex] is in thousands.
Next, we calculate the maximum profit by substituting this x-coordinate back into the profit function [tex]\( P(x) \)[/tex]:
[tex]\[ P(2) = -4(2)^2 + 16(2) - 7 \][/tex]
[tex]\[ P(2) = -4 \times 4 + 16 \times 2 - 7 \][/tex]
[tex]\[ P(2) = -16 + 32 - 7 \][/tex]
[tex]\[ P(2) = 16 - 7 \][/tex]
[tex]\[ P(2) = 9 \][/tex]
Hence, the maximum profit is [tex]\( 9 \)[/tex] thousand dollars, which means [tex]$9,000.
Therefore, the maximum profit is $[/tex]\[tex]$9,000$[/tex].
