Positive Parabola With No X-Axis Intersection: Evaluating the Always Decreasing Claim

Let's solve this problem step-by-step:

The problem addresses a quadratic function or "parabola" with the equation typically given as $f(x) = ax^2 + bx + c$, where $a, b,$ and $c$ are real numbers. For a parabola that opens upwards, the coefficient $a$ must be positive ($a > 0$).

Given that the parabola does not intersect or touch the x-axis, this means there are no real roots. This condition is satisfied when the discriminant $\Delta = b^2 - 4ac$ is less than zero ($\Delta < 0$).

For a parabola described by $ax^2 + bx + c$:

• The vertex forms the minimum point since $a > 0$. The vertex $(h, k)$ is calculated using $h = -\frac{b}{2a}$.
• The function is decreasing on the interval $(-\infty, h)$ and increasing on the interval $(h, \infty)$.

For it to be always decreasing would mean that $(-\infty, \infty)$, but we identified that it decreases up to $h$ and then increases beyond $h$. Thus, it cannot be always decreasing.

Conclusively, the statement that a positive parabola that does not intersect or touch the x-axis is always decreasing is False.