over which interval is the graph of $f(x)=\frac{1}{2}x^{2}+5x + 6$ increasing?\n(-6.5, ∞)\n(-5, ∞)\n(-∞…

Answer:

B. $(-5,\infty)$

Explanation:

Step1: Identify the function type

The function $f(x)=\frac{1}{2}x^{2}+5x + 6$ is a quadratic function in the form $y = ax^{2}+bx + c$, where $a=\frac{1}{2}$, $b = 5$, $c=6$.

Step2: Find the axis - of - symmetry

The formula for the axis of symmetry of a quadratic function $y=ax^{2}+bx + c$ is $x=-\frac{b}{2a}$. Substitute $a=\frac{1}{2}$ and $b = 5$ into the formula: $x=-\frac{5}{2\times\frac{1}{2}}=- 5$.

Step3: Determine the increasing interval

Since $a=\frac{1}{2}>0$, the parabola opens upward. For a parabola that opens upward, the function is increasing to the right of the axis of symmetry. So the function $f(x)$ is increasing on the interval $(-5,\infty)$.