Given the function:
[tex]g(x)=2x^2-12x+19[/tex]Let's determine if the function has a minimum or maximum.
The minimum and maximum of a function are the smallest and largest value of a function in a given range or domain
The given function has a minimum.
Apply the general equation of a quadratic function:
[tex]y=ax^2+bx+c[/tex]To find the minimum value, apply the formula:
[tex]x=-\frac{b}{2a}[/tex]Where:
b = -12
a = 2
Thus, we have:
[tex]\begin{gathered} x=-\frac{-12}{2(2)} \\ \\ x=-\frac{-12}{4} \\ \\ x=3 \end{gathered}[/tex]To find the function's minimum value, find f(3).
Substitute 3 for x in the function and evaluate:
[tex]\begin{gathered} f(x)=2x^2-12x+19 \\ \\ f(3)=2(3)^2-12(3)+19 \\ \\ f(3)=2(9)-36+19 \\ \\ f(3)=18-36+19 \\ \\ f(3)=1 \end{gathered}[/tex]Therefore, the function's minimum value is 1
Therefore, the functions minimum value occurs at:
x = 3
ANSWER:
• The function has a minimum
• Minimum value: 1
• The minimum occurs at: x = 3
