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Chapter 2: Problem 34

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. $$f(x)=x^{3}-4 x^{2}+2 ; \text { between } 0 \text { and } 1$$

Short Answer

Expert verified
By using the Intermediate Value Theorem, it can be proven that the given polynomial \(f(x) = x^3 - 4x^2 + 2\) has a real root between 0 and 1 since a change of sign was observed between these limits.

Step by step solution

01

Evaluate the function at the endpoints

Calculate the value of the function at the endpoints of the interval. That is, find \(f(0)\) and \(f(1)\) using the equation \(f(x) = x^3 - 4x^2 + 2\).
02

Determine the change of sign

With the values of \(f(0)\) and \(f(1)\), the next step is to evaluate if there is a change of sign between these two function values. If the function changes sign somewhere between \(f(0)\) and \(f(1)\), it means that it crosses the \(x\)-axis between 0 and 1.
03

Apply the Intermediate Value Theorem

If there is indeed a sign change between \(f(0)\) and \(f(1)\), apply the Intermediate Value Theorem to definitively conclude that there is a root of the function within the interval [0, 1].

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can never cross a vertical asymptote.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm solving a polynomial inequality that has a value for which the polynomial function is undefined.

a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\). b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\).

If you are given the equation of a rational function, explain how to find the horizontal asymptote, if there is one, of the function's graph.

a. Find the slant asymptote of the graph of each rational function and \(\mathbf{b} .\) Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$f(x)=\frac{x^{2}-4}{x}$$
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