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Chapter 3: Problem 11

Determine whether Rolle's Theorem can be applied to \(f\) on the closed interval \([a, b] .\) If Rolle's Theorem can be applied, find all values of \(c\) in the open interval \((a, b)\) such that \(f^{\prime}(c)=0\). $$f(x)=x^{2}-2 x,[0,2]$$

Short Answer

Expert verified
Rolle's theorem can be applied and the value of \(c\) in the open interval (0, 2) such that \(f^{\prime}(c)=0\) is \(c = 1\).

Step by step solution

01

Verifying Conditions

Check if the function \(f(x)=x^{2}-2 x\) is continuous on the closed interval [0, 2] and differentiable on the open interval (0, 2). The function is a polynomial which is continuous and differentiable on the entire real line, so it is continuous on [0, 2] and differentiable on (0, 2). Next, check if the condition f(a) = f(b) is met. Here, a=0 and b=2, so evaluate \(f(0)\) and \(f(2)\) . \(f(0)= (0)^{2}-2(0)=0\) and \(f(2)= (2)^{2}-2(2)=0\). Since \(f(0)=f(2)\), all the conditions of Rolle's theorem are met.
02

Differentiate the Function

Apply the power rule for differentiation to obtain the derivative of the function \(f(x) = x^2 - 2x\). This gives \(f'(x) = 2x - 2\)
03

Find \(c\)

Set the derivative equal to zero to find the value of \(c\) that satisfies the conclusion of Rolle's theorem: \(2x - 2 = 0\). Solve this to find \(x = 1\). Since 1 is in the open interval (0, 2), \(c = 1\) is the value we are looking for.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Function
A continuous function is one where small changes in the input result in small changes in the output. In simpler terms, the graph of a continuous function can be drawn without lifting the pencil off the paper. This is an essential feature for many calculus theorems, including Rolle's Theorem, which relies on functions being continuous on a closed interval
  • To verify continuity, check the endpoints and ensure the function does not have any breaks or jumps between them.
  • For example, while verifying Rolle's Theorem for the function \(f(x) = x^2 - 2x\) on the interval \([0, 2]\), we observe that it is a polynomial, hence continuous across all real numbers, including the interval \([0, 2]\).
This means from \(x = 0\) to \(x = 2\), the function behaves predictably without disruption.
Differentiable Function
Differentiability extends the idea of a function being smooth. A function is differentiable at a point if it has a defined derivative there, meaning it has a tangent line and no sharp corners or discontinuities. For Rolle's Theorem, this applies on an open interval and checks whether the function can be differentiated.
  • For a function to be differentiable over an interval (\((a, b)\)), it must be continuous over that interval and not contain any sharp corners or cusps.
  • The derivative tells you the slope of the tangent line to the function at any point within its domain.
In the case of \(f(x) = x^2 - 2x\), since it is a polynomial function, it is differentiable everywhere on the real line, including the interval \((0, 2)\). This smoothness allows us to easily find the derivative, \(f'(x) = 2x - 2\), a critical step in applying Rolle's theorem successfully.
Polynomial Function
A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. Specifically, it has the form \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\) where \(a_n, a_{n-1}, ..., a_0\) are constants.
  • Polynomials are some of the simplest and most well-understood functions.
  • They exhibit continuous and differentiable behavior across all real numbers, making them ideal candidates for Rolle's Theorem.
  • The function in question, \(f(x) = x^2 - 2x\), is a second-degree polynomial.
Because of their straightforward nature, the behavior of polynomial functions is predictable and easily examined, ensuring they meet the requirements for the theorem's application without complications.

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Most popular questions from this chapter

Modeling Data The table shows the world record times for running 1 mile, where \(t\) represents the year, with \(t=0\) corresponding to 1900 , and \(y\) is the time in minutes and seconds. \(y=\frac{3.351 t^{2}+42.461 t-543.730}{t^{2}}\) where the seconds have been changed to decimal parts of a minute. (a) Use a graphing utility to plot the data and graph the model. (b) Does there appear to be a limiting time for running 1 mile? Explain.

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Diminishing Returns The profit \(P\) (in thousands of dollars) for a company spending an amount \(s\) (in thousands of dollars) on advertising is \(P=-\frac{1}{10} s^{3}+6 s^{2}+400\) (a) Find the amount of money the company should spend on advertising in order to yield a maximum profit. (b) The point of diminishing returns is the point at which the rate of growth of the profit function begins to decline. Find the point of diminishing returns.

Consider \(\lim _{x \rightarrow \infty} \frac{3 x}{\sqrt{x^{2}+3}} .\) Use the definition of limits at infinity to find values of \(M\) that correspond to (a) \(\varepsilon=0.5\) and (b) \(\varepsilon=0.1\).
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