Look at the following function:
Determine for which values of the following is true:
f\left(x\right)>0
Look at the following function:
Determine for which values of the following is true:
f\left(x\right)>0
The function represents a parabola opening upwards since its leading coefficient is positive. Our task is to determine when the function is positive.
First, let's find the vertex of this parabola, which occurs at .
Here, and , so:
\begin{align*} x_{vertex} &= -\frac{-6}{2 \times 1} \\ &= \frac{6}{2} \\ &= 3. \end{align*}Next, we evaluate the function at this vertex:
\begin{align*} f(3) &= 3^2 - 6 \cdot 3 + 10 \\ &= 9 - 18 + 10 \\ &= 1. \end{align*}Since , which is greater than zero, we observe that at the vertex the function is indeed positive.
Moreover, because the parabola opens upwards and the vertex value is positive, the entire parabola lies above the x-axis. Consequently, for all .
Therefore, the function is positive for all values of .
The function is positive for all values of .