Find Intervals of Increase and Decrease: y = x² + 9x + 18

To find the intervals of increase and decrease for $y = x^2 + 9x + 18$, we'll follow these steps:

• Step 1: Differentiate the function.
• Step 2: Find critical points by setting the derivative equal to zero.
• Step 3: Determine the sign of the derivative in each interval.
• Step 4: Interpret these signs to define intervals of increase and decrease.

First, let's find the first derivative of our function. The given function is $y = x^2 + 9x + 18$.

The derivative is calculated as follows:
$f'(x) = \frac{d}{dx}(x^2 + 9x + 18) = 2x + 9$.

Next, set $f'(x)$ to zero to find the critical points:
$2x + 9 = 0$.

Solve for $x$:
$2x = -9$
$x = -\frac{9}{2}$ or $x = -4.5$.

Now, we determine the sign of $f'(x)$ in intervals determined by this critical point: test on either side of $x = -4.5$.

• For $x < -4.5$, select $x = -5$: $f'(-5) = 2(-5) + 9 = -10 + 9 = -1$. As $f'(-5) < 0$, the function is decreasing.
• For $x > -4.5$, select $x = 0$: $f'(0) = 2(0) + 9 = 9$. As $f'(0) > 0$, the function is increasing.

This analysis reveals:

• The function is decreasing on the interval $x < -4.5$.
• The function is increasing on the interval $x > -4.5$.

Therefore, the final answer is:
$\searrow~:x < -4\frac{1}{2}~~ \\ \nearrow~:x > -4\frac{1}{2}$.