Find Increasing and Decreasing Intervals: y = 2x² - 5x + 3

To solve this problem, we'll start by differentiating the function:

The given function is $y = 2x^2 - 5x + 3$. To find intervals of increase and decrease, we need the first derivative $y'$:

$y' = \frac{d}{dx}(2x^2 - 5x + 3) = 4x - 5$.

Next, find the critical points by setting $y' = 0$ and solving for $x$:

$4x - 5 = 0$
$4x = 5$
$x = \frac{5}{4}$.

The critical point is at $x = \frac{5}{4}$, which is $x = 1\frac{1}{4}$ in mixed fraction form.

Now, determine where the function is increasing or decreasing by analyzing the sign of $y' = 4x - 5$:

• For $x < \frac{5}{4}$, $y' = 4x - 5 < 0$, indicating the function is decreasing.
• For $x > \frac{5}{4}$, $y' = 4x - 5 > 0$, indicating the function is increasing.

Therefore, the intervals of increase and decrease are:

• Decreasing for $x < 1\frac{1}{4}$.
• Increasing for $x > 1\frac{1}{4}$.

The correct answer is the interval description:

$\searrow~:x < 1\frac{1}{4}~~ \\ \nearrow~:x > 1\frac{1}{4}$.