To determine over which interval the function [tex]\( f(x) = \frac{1}{2} x^2 + 5x + 6 \)[/tex] is increasing, we need to analyze the behavior of the derivative of the function.
1. Find the derivative of the function:
The first step is to compute the derivative [tex]\( f'(x) \)[/tex] of the function [tex]\( f(x) \)[/tex].
[tex]\[
f(x) = \frac{1}{2} x^2 + 5x + 6
\][/tex]
Using the power rule for differentiation, we get:
[tex]\[
f'(x) = \frac{d}{dx} \left( \frac{1}{2} x^2 + 5x + 6 \right) = \frac{1}{2} \cdot 2x + 5 = x + 5
\][/tex]
2. Set the derivative greater than zero to find the increasing intervals:
A function is increasing where its derivative is positive. Thus, we set up the inequality:
[tex]\[
f'(x) > 0
\][/tex]
Substituting [tex]\( f'(x) = x + 5 \)[/tex]:
[tex]\[
x + 5 > 0
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x > -5
\][/tex]
3. Determine the correct interval:
The inequality [tex]\( x > -5 \)[/tex] tells us that the function [tex]\( f(x) \)[/tex] is increasing for all [tex]\( x \)[/tex] values greater than [tex]\( -5 \)[/tex]. This corresponds to the interval:
[tex]\[
(-5, \infty)
\][/tex]
Thus, the interval over which the graph of [tex]\( f(x) = \frac{1}{2} x^2 + 5x + 6 \)[/tex] is increasing is:
[tex]\[
(-5, \infty)
\][/tex]
This matches one of the provided choices:
[tex]\[
\boxed{(-5, \infty)}
\][/tex]
