Determine the Positive and Negative Domains of y = 1/3x² + 1/2x + 2/3

To solve this problem, we need to analyze the function $y = \frac{1}{3}x^2 + \frac{1}{2}x + \frac{2}{3}$ to determine where it is positive or negative. This function is quadratic, so it is a parabola. Let us find the domain of positive and negative values.

First, let's determine where the function is zero by finding its roots. This involves using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, we have $a = \frac{1}{3}$, $b = \frac{1}{2}$, and $c = \frac{2}{3}$. The discriminant is calculated as follows:

$b^2 - 4ac = \left(\frac{1}{2}\right)^2 - 4 \cdot \frac{1}{3} \cdot \frac{2}{3} = \frac{1}{4} - \frac{8}{9}$

Calculating gives:

$\frac{1}{4} \approx 0.25$ and $\frac{8}{9} \approx 0.888$ which results in:

$0.25 - 0.888 = -0.638$

Since the discriminant is negative, there are no real roots. As the parabola opens upwards (since $a = \frac{1}{3} > 0$), the function never crosses the x-axis. Therefore, the function remains positive for all real values of $x$ and is never negative.

Thus, the positive domain consists of all real numbers, while there is no negative domain:

$x > 0$: \) for all $x$

$x < 0$: \) none