If function f has exactly two distinct real zeros, which of the following graphs could be the complete graph of f (x) ?

Options

A.
B.
C.
D.

Answer

To solve the problem of identifying which graph could represent a function \( f(x) \) with exactly two distinct real zeros, we need to analyze the characteristics of the graphs provided. Here’s a step-by-step solution:

### Step 1: Understand the concept of real zeros
Real zeros of a function \( f(x) \) are the points where the graph intersects the x-axis. If a function has exactly two distinct real zeros, it means the graph must touch or cross the x-axis at exactly two different points.

**Hint:** Look for points where the graph intersects the x-axis; these points represent the zeros of the function.

### Step 2: Analyze the first graph
In the first graph, observe the points where the graph touches the x-axis. If the graph intersects the x-axis at two distinct points, then this graph could represent the function \( f(x) \).

**Hint:** Count the number of times the graph crosses or touches the x-axis.

### Step 3: Analyze the second graph
In the second graph, check how many times the graph intersects the x-axis. If it touches the x-axis at three different points, then it has three distinct real zeros, which does not satisfy the condition of having exactly two distinct real zeros.

**Hint:** If the graph intersects the x-axis more than two times, it cannot be the correct option.

### Step 4: Analyze the third graph
In the third graph, if the graph only touches the x-axis at one point, it indicates that there is only one real zero (with multiplicity). This does not meet the requirement of having two distinct real zeros.

**Hint:** A single touch at the x-axis indicates a repeated root, not distinct zeros.

### Step 5: Analyze the fourth graph
In the fourth graph, if the graph intersects the x-axis at three points, similar to the second graph, it indicates three distinct real zeros. Therefore, this option is also not valid.

**Hint:** Again, look for the number of distinct intersections with the x-axis.

### Conclusion
After analyzing all four graphs, the only graph that meets the requirement of having exactly two distinct real zeros is the first graph. Therefore, the answer is:

**The complete graph of \( f(x) \) is represented by option one.**