Find Intervals of Increase and Decrease for y = (5x - 1)²

To find the intervals of increase and decrease for the function $y = (5x - 1)^2$, follow these steps:

• Step 1: Differentiate the function $y = (5x - 1)^2$.
• Step 2: Set the derivative equal to zero to find critical points.
• Step 3: Test intervals around the critical point to determine where the function is increasing or decreasing.

Now, let's work through each step.

Step 1: Differentiate the function.
The given function is $y = (5x - 1)^2$. Using the chain rule, the derivative $y'$ is:

Step 2: Find critical points.
Set $y' = 0$:

Solving for $x$, we get:

Step 3: Determine intervals of increase and decrease.
Test intervals around the critical point $x = \frac{1}{5}$.

• For $x < \frac{1}{5}$ (e.g., $x = 0$), $y' = 50(0) - 10 = -10$. Therefore, $y' < 0$, and the function is decreasing.
• For $x > \frac{1}{5}$ (e.g., $x = 1$), $y' = 50(1) - 10 = 40$. Therefore, $y' > 0$, and the function is increasing.

Thus, the function is decreasing on the interval $x < \frac{1}{5}$ and increasing on the interval $x > \frac{1}{5}$.

Therefore, the correct choice is:

$\searrow: x < \frac{1}{5} \\ \nearrow: x > \frac{1}{5}$