The image you've uploaded contains a mathematical problem related to a quadratic function. It asks you to evaluate the correctness of various inequalities based on the graph of the function.

The function is given as: $y = f(x) = ax^2 + bx + c$ and you're asked to examine the following statements:

1. $f(2) > 0$
2. $f(x) > 0 \Leftrightarrow x \in (0; 1) \cup (4; \infty)$
3. $f(x) \leq 0 \Leftrightarrow x \in [1; 4]$
4. $f(x) = 0$ has 2 distinct real roots

Let me now carefully analyze the graph and the statements to help you determine which are true or false.

Step 1: Look at the Graph's Key Features

From the graph, we can extract the following information:

• The vertex of the parabola appears to be at $(1, 4)$, suggesting the function has a maximum point at $x = 1$ and the parabola opens downward.
• The function crosses the x-axis at two points, likely between $x = 0$ and $x = 4$, indicating two real roots.

Step 2: Analyzing the Statements

Statement a: $f(2) > 0$

• We observe that at $x = 2$, the function is below the x-axis, so $f(2) < 0$. Therefore, this statement is false.

Statement b: $f(x) > 0 \Leftrightarrow x \in (0; 1) \cup (4; \infty)$

• The parabola is above the x-axis in the intervals $(0; 1)$ and $(4; \infty)$. This statement is true.

Statement c: $f(x) \leq 0 \Leftrightarrow x \in [1; 4]$

• The function is non-positive between the roots (around $x = 1$ to $x = 4$), meaning this statement is true.

Statement d: $f(x) = 0$ has 2 distinct real roots

• The graph shows that the function crosses the x-axis at two distinct points. Hence, this statement is true.

Conclusion:

• a) False
• b) True
• c) True
• d) True

Would you like to dive into more details for any specific part of this solution, or do you have other questions?

Here are some related questions you could explore:

1. How do you determine the vertex of a quadratic function from its graph?
2. What role does the discriminant of a quadratic equation play in determining the number of real roots?
3. How do you solve inequalities involving quadratic functions?
4. How can you find the exact values of the roots from the graph of a parabola?
5. How does the orientation of a parabola (upward or downward) affect its solutions?

Tip: To find the roots of a quadratic function, you can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where the discriminant $\Delta = b^2 - 4ac$ determines the nature of the roots.