Determine whether there is a maximum or minimum value for the given function, and find that value.
-$$f ( x ) = x ^ { 2 } + 2 x - 9$$
A) Maximum: 10
B) Minimum: $- 10$
C) Maximum: $- 10$
D) Minimum: 0

Question

Quizplus · Accepted Answer

The given function $f(x) = x^2 + 2x - 9$ is a quadratic function with a positive leading coefficient (1 in this case), which means its graph is a parabola opening upwards. Therefore, it has a minimum value but no maximum. To find the minimum value, we can complete the square or use the vertex formula. The vertex of a parabola given by $y = ax^2 + bx + c$ is at $(-b/2a, f(-b/2a))$. Here, $a = 1$ and $b = 2$, so the x-coordinate of the vertex is $-2/(2*1) = -1$. Plugging this back into the function gives $f(-1) = (-1)^2 + 2(-1) - 9 = 1 - 2 - 9 = -10$. Therefore, the minimum value of the function is $-10$.

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