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Chapter 2: Problem 57

a. Use the Intermediate Value Theorem to show that the following equations have a solution on the given interval. b. Use a graphing utility to find all the solutions to the equation on the given interval. c. Illustrate your answers with an appropriate graph. $$x^{3}-5 x^{2}+2 x=-1 ;(-1,5)$$

Short Answer

Expert verified
Short Answer: To prove that the given function $$f(x) = x^3 - 5x^2 + 2x -1$$ has at least one solution on the interval \((-1, 5)\), we can apply the Intermediate Value Theorem (IVT). Since the function is a polynomial, it is continuous everywhere, including the interval [-1, 5]. We find f(-1) = -9 and f(5) = 9. Since f(-1) < 0 and f(5) > 0, there exists at least one solution in the given interval by the IVT. To find all the solutions and create an appropriate graph, use a graphing utility (such as Desmos, Grapher, or a graphing calculator) to identify the x-intercepts within the interval (-1, 5).

Step by step solution

01

Intermediate Value Theorem (IVT)

First, we need to prove that the given function has at least one solution on the interval (-1,5) using the Intermediate Value Theorem. The IVT states that if a function is continuous on a closed interval [a, b], and N is any real number between f(a) and f(b), then there exists at least one c in the interval (a, b) such that f(c) = N. Let's check if our function is continuous. Since it's a polynomial function, it's continuous everywhere. Therefore, our function f(x) is continuous on the interval [-1, 5]. Next, let's compute f(-1) and f(5): f(-1) = \((-1)^3 - 5(-1)^2 + 2(-1) - 1 = -1 - 5 - 2 - 1 = -9\) f(5) = \((5)^3 - 5(5)^2 + 2(5) - 1 = 125 - 125 + 10 - 1 = 9\) Since f(-1) < 0 and f(5) > 0 and N = 0 is a number between f(-1) and f(5), there exists at least one solution to the equation f(x) = 0 on the interval (-1,5) by the IVT.
02

Using a Graphing Utility (for tasks b and c)

To find all the solutions and create a proper graph to illustrate our answer, we'll use a graphing utility (such as Desmos, Grapher, or a graphing calculator). For task b, plug the function f(x) = \(x^3 - 5x^2 + 2x -1\) into the graphing utility and find the x-intercepts (where it crosses the x-axis) within the interval (-1, 5). These x-intercepts indicate the solutions to the equation in the given interval. For task c, create an appropriate graph using the same graphing utility. Ensure that the graph showcases the function in the given interval, with its x-intercepts clearly visible.

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Most popular questions from this chapter

Prove the following statements to establish the fact that \(\lim _{x \rightarrow a} f(x)=L\) if and only if \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\) a. If \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L,\) then \(\lim _{x \rightarrow a} f(x)=L\) b. If \(\lim _{x \rightarrow a} f(x)=L,\) then \(\lim _{x \rightarrow a^{-}} f(x)=L\) and \(\lim _{x \rightarrow a^{+}} f(x)=L\)

If a function \(f\) represents a system that varies in time, the existence of \(\lim _{t \rightarrow \infty} f(t)\) means that the system reaches a steady state (or equilibrium). For the following systems, determine if a steady state exists and give the steady-state value. The amount of drug (in milligrams) in the blood after an IV tube is inserted is \(m(t)=200\left(1-2^{-t}\right)\).

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