True or False. The graph of f(x)=x^3-x^2-2 crosses the x -axis between x=1 and x=2 . Justify you

step 1 Okay, so this is a standardized test question, and we're asked, does this function cross the x -

True or False. The graph of f(x)=x^3-x^2-2 crosses the x...

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Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem states that any continuous function that takes on values of opposite signs at two points must take on the value zero at some point in between. This theorem is fundamental in proving the existence of roots within an interval, as it guarantees that a crossing of the x-axis occurs if the function changes sign.

Continuity

Continuity is the property of a function where small changes in the input result in small changes in the output, with no abrupt jumps or breaks. This property is particularly important for applying the Intermediate Value Theorem, as the theorem relies on the function being continuous over the interval in question.

Polynomial Functions

Polynomial functions are expressions constructed from variables raised to non-negative integer powers and coefficients combined by addition, subtraction, and multiplication. These functions are inherently continuous everywhere in their domain, which makes them suitable for analysis using the Intermediate Value Theorem to locate roots or intersections with the x-axis.

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