Find Intervals of Increase and Decrease: Analyzing y = (x + 1/6)(-x - 4 1/9)

To find the intervals of increase and decrease for the function $y = \left(x + \frac{1}{6}\right)\left(-x - 4\frac{1}{9}\right)$, we follow these steps:

• Step 1: Expand the function. This means multiplying out the terms:
  $y = (x + \frac{1}{6})(-x - \frac{37}{9}) = -x^2 - \frac{37}{9}x - \frac{1}{6}x - \frac{37}{54}$
  Simplifying gives $y = -x^2 - \frac{223}{54}x - \frac{37}{54}$.
• Step 2: Find the derivative of the function.
  $\frac{dy}{dx} = -2x - \frac{223}{54}$.
• Step 3: Find critical points by setting the derivative to zero:
  $-2x - \frac{223}{54} = 0$. Solving for $x$ gives $x = -\frac{223}{108}$, which simplifies to $x = -\frac{77}{36}$.
• Step 4: Determine intervals of increase and decrease by testing values around the critical point $x = -\frac{77}{36}$.
  If $x \lt -\frac{77}{36}$, the derivative $-2x - \frac{223}{54}$ is positive, indicating an increasing interval.
  If $x \gt -\frac{77}{36}$, the derivative $-2x - \frac{223}{54}$ is negative, indicating a decreasing interval.

Thus, the function is increasing on the interval $x \lt -\frac{77}{36}$ and decreasing on the interval $x \gt -\frac{77}{36}$.

Therefore, the intervals of increase and decrease are:
$\searrow:x \gt -\frac{77}{36} \\ \nearrow:x \lt -\frac{77}{36}$