Let's break down the given quadratic function and answer the required parts step by step.

The function given is:

$f(x) = -2x^2 + 2x - 1$

This is a quadratic function with a downward-opening parabola (because the coefficient of $x^2$ is negative).

Finding the Vertex

The vertex form of a quadratic function is given by:

$x = \frac{-b}{2a}$

For the given function $f(x) = -2x^2 + 2x - 1$:

• $a = -2$
• $b = 2$

Substitute the values into the vertex formula:

$x = \frac{-2}{2(-2)} = \frac{-2}{-4} = \frac{1}{2}$

So, the x-coordinate of the vertex is $x = \frac{1}{2}$.

(a) Increasing Interval:

Since the parabola opens downward, the function is increasing to the left of the vertex and decreasing to the right. Therefore, the function is increasing on the interval:

$\left( -\infty, \frac{1}{2} \right]$

(b) Decreasing Interval:

The function decreases after the vertex. Therefore, the function is decreasing on the interval:

$\left[ \frac{1}{2}, \infty \right)$

(f) Determining where $f(x) > 0$ and $f(x) < 0$:

To find where the function is positive or negative, we need to solve for the roots of the quadratic equation by using the quadratic formula:

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

Substituting $a = -2$, $b = 2$, and $c = -1$ into the quadratic formula:

$x = \frac{-2 \pm \sqrt{2^2 - 4(-2)(-1)}}{2(-2)} = \frac{-2 \pm \sqrt{4 - 8}}{-4} = \frac{-2 \pm \sqrt{-4}}{-4}$

Since the discriminant is negative ($\sqrt{-4} = 2i$), this quadratic has no real roots. Therefore, the parabola does not cross the x-axis, and since the parabola opens downward, the function is always negative.

Thus, the correct choice for part (f) is:

C. $f(x) > 0$ on an empty set and $f(x)$ is never positive.

Would you like more details or have any questions on these steps? Here are five related questions to expand your understanding:

1. How can the discriminant help determine the number of real roots for a quadratic equation?
2. How does the leading coefficient affect the direction of a parabola?
3. Can you describe the relationship between the vertex and the intervals of increasing and decreasing behavior for a quadratic function?
4. What are the implications of a quadratic function having no real roots?
5. How do you rewrite a quadratic equation in vertex form?

Tip: When solving quadratic equations, always check the discriminant ($b^2 - 4ac$) to determine whether the solutions are real or complex.