What can be inferred about the roots of a cubic function based on its graph?

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Question: What can be inferred about the roots of a cubic function based on its graph?

Options:

Correct Answer: It must have at least one real root.

Solution:

A cubic function must have at least one real root due to the Intermediate Value Theorem, as it is a continuous function.

What can be inferred about the roots of a cubic function based on its graph?

What can be inferred about the roots of a cubic function based on its graph?

Step 1: Understand what a cubic function is. A cubic function is a mathematical expression of the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a is not zero.
Step 2: Know that the graph of a cubic function is a continuous curve. This means there are no breaks or gaps in the graph.
Step 3: Learn about the Intermediate Value Theorem. This theorem states that if a function is continuous on an interval and takes on different values at the endpoints of that interval, then it must cross the x-axis at least once within that interval.
Step 4: Apply the Intermediate Value Theorem to cubic functions. Since a cubic function is continuous, it must cross the x-axis at least once, which means it has at least one real root.
Step 5: Recognize that a cubic function can have up to three real roots. The graph can touch or cross the x-axis at one, two, or three points, depending on the specific function.

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