Determine whether there is a maximum or minimum value for the given function, and find that value.
-$$f ( x ) = x ^ { 2 } - 7 x + 6$$
A) Minimum: $- \frac { 25 } { 4 }$
B) Minimum: 11
C) Maximum: $\frac { 171 } { 4 }$
D) Minimum: $\frac { 171 } { 4 }$

Question

Quizplus · Accepted Answer

The function $f(x) = x^2 - 7x + 6$ is a quadratic function with a positive leading coefficient, indicating it opens upwards and thus has a minimum value. The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The x-coordinate of the vertex can be found using the formula $h = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients from the quadratic equation in the form $ax^2 + bx + c$. For this function, $a = 1$ and $b = -7$, so $h = -\frac{-7}{2(1)} = \frac{7}{2}$. Substituting $x = \frac{7}{2}$ back into the original equation gives the minimum value: $f\left(\frac{7}{2}\right) = \left(\frac{7}{2}\right)^2 - 7\left(\frac{7}{2}\right) + 6 = \frac{49}{4} - \frac{49}{2} + 6 = \frac{49}{4} - \frac{98}{4} + \frac{24}{4} = -\frac{25}{4}$.

Determine Whether There Is a Maximum or Minimum Value for the Given