Determine whether the given quadratic function has a minimum value or maximum value. Then find the
coordinates of the minimum or maximum point.
-$$f ( x ) = 4 x ^ { 2 } + 2 x + 2$$
A) minimum; $\left( - \frac { 1 } { 4 } , \frac { 7 } { 4 } \right)$
B) maximum; $\left( - \frac { 1 } { 4 } , \frac { 7 } { 4 } \right)$
C) minimum; $\left( \frac { 7 } { 4 } , - \frac { 1 } { 4 } \right)$
D) maximum; $\left( \frac { 7 } { 4 } , - \frac { 1 } { 4 } \right)$

Question

Quizplus · Accepted Answer

The coefficient of $x^2$ in the quadratic equation $f(x) = 4x^2 + 2x + 2$ is positive, indicating the parabola opens upwards and thus has a minimum value. The vertex (minimum point) of the parabola can be found using the formula $-\frac{b}{2a}$ for the x-coordinate, where $a = 4$ and $b = 2$, giving $-\frac{2}{2*4} = -\frac{1}{4}$. Substituting $-\frac{1}{4}$ back into the equation for $f(x)$ gives the y-coordinate as $\frac{7}{4}$. Therefore, the correct answer is minimum; $\left( - \frac { 1 } { 4 } , \frac { 7 } { 4 } \right)$.

Determine Whether the Given Quadratic Function Has a Minimum Value f(x)=4x2+2x+2f ( x ) = 4 x ^ { 2 } + 2 x + 2f(x)=4x2+2x+2

Determine Whether the Given Quadratic Function Has a Minimum Value $f ( x ) = 4 x ^ { 2 } + 2 x + 2$