The image shows a problem involving a quadratic function and its graph. The problem asks to find the values of $x$ that satisfy the inequality $f(x) \geq 0$, where $f(x) = -x(x-2)$.

Step-by-Step Solution

1. Identify the function: The function given is $f(x) = -x(x-2)$. Expanding this, we have: $f(x) = -x^2 + 2x$ This is a quadratic equation of the form $ax^2 + bx + c$ where $a = -1$, $b = 2$, and $c = 0$.

2. Analyze the graph: The graph shows a downward-opening parabola, which makes sense because the coefficient of $x^2$ is negative ($a = -1$).

3. Find the roots: The roots of the quadratic equation are the points where $f(x) = 0$. We can find the roots by solving: $-x(x-2) = 0$ This gives: $x = 0 \quad \text{or} \quad x = 2$ So, the parabola crosses the x-axis at $x = 0$ and $x = 2$.

4. Determine the intervals for $f(x) \geq 0$: The parabola is above the x-axis (i.e., $f(x) \geq 0$) between the roots $x = 0$ and $x = 2$. Therefore, the solution to the inequality $f(x) \geq 0$ is: $0 \leq x \leq 2$

Final Answer

The correct values of $x$ that satisfy $f(x) \geq 0$ are in the interval $[0, 2]$.

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Relative Questions

1. How do you solve quadratic inequalities in general?
2. What is the significance of the vertex in the graph of a quadratic function?
3. How can you determine whether a parabola opens upwards or downwards?
4. What are the different methods to find the roots of a quadratic equation?
5. How do you graph a quadratic function by hand?

Tip

When dealing with quadratic inequalities, always remember to identify the roots and check the sign of the quadratic expression in the intervals determined by the roots.