Determine whether the quadratic function shown below has a minimum or maximum, then determine the minimum or maximum value of the function.
f(x)=(x−4)(x−2)
The quadratic function f(x)=(x−4)(x−2) is in the form of f(x)=ax^2+bx+c, where a=1, b=-6, and c=8.
To determine whether the function has a minimum or maximum, we can look at the coefficient of the x^2 term, which is a. Since a is positive (a=1), the quadratic function will have a minimum.
To determine the minimum value of the function, we can find the vertex of the parabola. The vertex can be found using the formula x = -b/2a.
In this case, x = -(-6)/(2*1) = 6/2 = 3.
To find the corresponding y-value of the vertex, we substitute x = 3 into the function f(x)=(x−4)(x−2).
f(3) = (3-4)(3-2) = (-1)(1) = -1.
Therefore, the quadratic function f(x)=(x−4)(x−2) has a minimum at the vertex (3, -1), and the minimum value of the function is -1.