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To find if this quadratic function has a minimum or a maximum value, we need to see the quadratic coefficient of the function. If the value of this coefficient is negative, the function will have a maximum. If the coefficient is positive, the quadratic function will have a minimum.
Therefore, in this case, we have:
[tex]g(x)=-x^2+10x-27[/tex]We can see that "in front of x^2, we have that the coefficient is negative. Therefore, this function has a maximum.
To find where this maximum occurs, we can use the formula for the vertex of a parabola (quadratic function):
[tex]x_v=-\frac{b}{2a},y_v=c-\frac{b^2}{4a}[/tex]We can get a, b, and c from the quadratic function in question:
[tex]ax^2+bx+c\Rightarrow-x^2+10x-27[/tex]Then, we have that:
a = -1
b = 10
c = -27
Then, to find the value of x when the maximum occurs, we have:
[tex]x_v=-\frac{10}{2(-1)}=-\frac{10}{-2}=5\Rightarrow x_v=5[/tex]And now, to find the value of y when the maximum occurs, we also have:
[tex]y_v=-27-\frac{(10)^2}{4\cdot(-1)}=-27-\frac{100}{-4}=-27+25=-2\Rightarrow y_v=-2[/tex]Therefore, the value for x is equal to 5 (x = 5), and the value for y is equal to -2 ( y = -2) when the quadratic function takes its maximum value.
A graph for this function can be seen as follows:
In summary, the function has a maximum value since the quadratic coefficient is negative, and the x-coordinate for this maximum is x = 5, and the y-coordinate for this maximum is y = -2 (and this is the maximum value for the function).
