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Chapter 1: Problem 65

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}+x-1, \quad[0,5], \quad f(c)=11 $$

Short Answer

Expert verified
The Intermediate Value Theorem applies to the function on the given interval, and the value of \(c\) guaranteed by the theorem in this case is \(c = 3\).

Step by step solution

01

Verifying Conditions of the Intermediate Value Theorem

First, consider the function \(f(x)=x^{2}+x-1\). It is a polynomial function, and polynomial functions are known to be continuous for all real numbers. Thus, \(f(x)\) is continuous on [0,5]. For the Intermediate Value Theorem to apply, k should be a number between \(f(a)\) and \(f(b)\), where [a,b] is the given interval. Here, \(a=0\), \(b=5\), and \(k=11\). Calculate \(f(a)\) and \(f(b)\) to check. \(f(a) = f(0) = 0^2 + 0 - 1 = -1\)\(f(b) = f(5) = 5^2 + 5 - 1 = 29\)Confirm that k is between \(f(a)\) and \(f(b)\), or in other words, \(f(a) < k < f(b)\). Here, \(-1 < 11 < 29\). Since this is true, the conditions of the Intermediate Value Theorem are met.
02

Finding the Value of C

To find the value of \(c\) that yields \(f(c) = 11\), solve the equation \(f(x) = 11\) for \(x\). This gives\(x^2 + x - 1 = 11\)Simplify this to \(x^2 + x - 12 = 0\)Now, use the quadratic formula, \(x = [-b \pm \sqrt{(b^2 - 4ac)}] / 2a\). In this case, \(a=1\), \(b=1\), and \(c=-12\). Solving this using the quadratic formula gives \(x = [-1 \pm \sqrt{(1 + 48)}] / 2\), which simplifies to \(x = [-1 \pm \sqrt{49}] / 2\). This yields two solutions: \(x = 3\) and \(x = -4\). Since the guaranteed \(c\) needs to be within the interval [0,5], \(c = 3\) is the only valid solution.

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Most popular questions from this chapter

Use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ h(\theta)=1+\theta-3 \tan \theta $$

Use the Intermediate Value Theorem and a graphing utility to approximate the zero of the function in the interval [0, 1]. Repeatedly "zoom in" on the graph of the function to approximate the zero accurate to two decimal places. Use the zero or root feature of the graphing utility to approximate the zero accurate to four decimal places. $$ g(t)=2 \cos t-3 t $$

In Exercises 117-126, write the expression in algebraic form. \(\tan (\arctan x)\)

In Exercises \(131-134,\) sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arcsin (x-1) $$

True or False? In Exercises \(50-53\), determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(p(x)\) is a polynomial, then the graph of the function given by \(f(x)=\frac{p(x)}{x-1}\) has a vertical asymptote at \(x=1\)
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