Answer:
A
Step-by-step explanation:
Given a parabola in standard form, y = ax² + bx + c ( a ≠ 0 ), then
minimum/ maximum value is the y- coordinate of the vertex.
The x- coordinate of the vertex is
[tex]x_{vertex}[/tex] = - [tex]\frac{b}{2a}[/tex]
y = x² - [tex]\frac{5}{3}[/tex] x + [tex]\frac{31}{36}[/tex] ← is in standard form
with a = 1 and b = - [tex]\frac{5}{3}[/tex] , thus
[tex]\frac{x}{vertex}[/tex] = - [tex]\frac{-\frac{5}{3} }{2}[/tex] = [tex]\frac{5}{6}[/tex]
Substitute this value into y
y = ([tex]\frac{5}{6}[/tex] )² - [tex]\frac{5}{3}[/tex] ([tex]\frac{5}{6}[/tex] ) + [tex]\frac{31}{36}[/tex]
= [tex]\frac{25}{36}[/tex] - [tex]\frac{25}{18}[/tex] + [tex]\frac{31}{36}[/tex] = [tex]\frac{1}{6}[/tex]
Since a > 1 then the vertex is a minimum, thus
minimum value = [tex]\frac{1}{6}[/tex] → A
