Determine whether the given quadratic function has a minimum value or maximum value. Then find the coordinates of
the minimum or maximum point.
-$$f ( x ) = 2 x ^ { 2 } - 2 x - 3$$
A) maximum; $\left( \frac { 1 } { 2 } , - \frac { 7 } { 2 } \right)$
B) minimum; $\left( \frac { 1 } { 2 } , - \frac { 7 } { 2 } \right)$
C) maximum; $\left( - \frac { 7 } { 2 } , \frac { 1 } { 2 } \right)$
D) minimum; $\left( - \frac { 7 } { 2 } , \frac { 1 } { 2 } \right)$

Question

Quizplus · Accepted Answer

The quadratic function $f(x) = 2x^2 - 2x - 3$ has a positive leading coefficient (2), indicating it opens upwards and thus has a minimum value. The vertex (minimum point) of a parabola given by $f(x) = ax^2 + bx + c$ can be found using the formula $x = -\frac{b}{2a}$. Here, $a = 2$ and $b = -2$, so $x = -\frac{-2}{2*2} = \frac{1}{2}$. Substituting $x = \frac{1}{2}$ into the original equation gives $f\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^2 - 2\left(\frac{1}{2}\right) - 3 = -\frac{7}{2}$. Therefore, the minimum point is $\left(\frac{1}{2}, -\frac{7}{2}\right)$.

Determine Whether the Given Quadratic Function Has a Minimum Value f(x)=2x2−2x−3f ( x ) = 2 x ^ { 2 } - 2 x - 3f(x)=2x2−2x−3

Determine Whether the Given Quadratic Function Has a Minimum Value $f ( x ) = 2 x ^ { 2 } - 2 x - 3$