Use the Intermediate value theorem to show that f has a zero between a and b f(x)=-x^4+3 x^3-2 x

step 1 In this question, we're given a function, FX is equal to minus X raised to power 4 plus 3x cube

Use the Intermediate value theorem to show that f has a zero...

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Key Concepts

Sign Change in Function Values

A sign change occurs when the function takes a positive value at one endpoint and a negative value at the other. This is an important concept because a sign change in a continuous function guarantees that the function must cross zero somewhere in between, thus ensuring the existence of a root.

Polynomial Functions

Polynomial functions are expressions consisting of variables and coefficients combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. They are inherently continuous and differentiable over the entire real line, making them ideal candidates for applying the Intermediate Value Theorem to locate zeros.

Intermediate Value Theorem

The Intermediate Value Theorem states that if a function is continuous on a closed interval [a, b] and takes on two different values at the endpoints, then it must take on every value between those two values at some point in the interval. This is key in proving the existence of a zero when the function values at the endpoints have opposite signs.

Continuity

Continuity means that a function does not have any breaks, jumps, or holes in its graph over a given interval. This property is crucial because the Intermediate Value Theorem applies only to continuous functions over a closed interval.

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