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Chapter 4: Problem 4

Use the Monotonicity Theorem to find where the given function is increasing and where it is decreasing. $$ f(x)=x^{3}-1 $$

Short Answer

Expert verified
The function is increasing everywhere in its domain.

Step by step solution

01

Find the Derivative

To determine where the function is increasing or decreasing, we first need to find its derivative. The derivative of the function \( f(x) = x^3 - 1 \) is \( f'(x) = 3x^2 \).
02

Find Critical Points

Critical points occur where the derivative is zero or undefined. Set \( f'(x) = 3x^2 = 0 \) and solve for \( x \). The solution is \( x = 0 \). There are no points where the derivative is undefined since it is a polynomial.
03

Determine Sign of Derivative

Examine the sign of \( f'(x) = 3x^2 \) to determine where the function is increasing or decreasing. Since \( 3x^2 \) is always non-negative and equals zero only at \( x = 0 \), \( f'(x) > 0 \) for all \( x eq 0 \).
04

Conclusion on Monotonicity

Since the derivative \( f'(x) = 3x^2 \) is positive for all \( x eq 0 \), the function \( f(x) = x^3 - 1 \) is increasing on the entire set of real numbers except at the point \( x = 0 \). Since it does not decrease and is constant only at this single point, it is effectively increasing everywhere.

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Key Concepts

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

Derivative
To understand the monotonicity of a function, calculating the derivative is crucial. The derivative provides a way to measure how a function changes as its input changes. For the function given, \( f(x) = x^3 - 1 \), its derivative is \( f'(x) = 3x^2 \). This involves applying basic differentiation rules: the power rule, where the derivative of \( x^n \) is \( nx^{n-1} \). Here, \( x^3 \) becomes \( 3x^2 \), showcasing how the derivative tells us the rate of change of \( f(x) \). The derivative is a key tool in analyzing a function's behavior.
Critical Points
Critical points are found where the derivative of a function is zero or undefined. These points are potential locations where the function could change its increasing or decreasing nature. For a polynomial like \( 3x^2 \), it鈥檚 critical to understand that the derivative is never undefined. Solving \( 3x^2 = 0 \) yields \( x = 0 \), pinpointing our critical point. This means \( x = 0 \) is a location where something notable happens in our function's slope, affecting its overall shape and behavior.
Polynomial
Polynomials are algebraic expressions consisting of variables and coefficients, involving terms with non-negative integer exponents. The function \( f(x) = x^3 - 1 \) is a polynomial of degree 3, where 'degree' signifies the highest power of the variable. The simplicity and continuity of polynomials make them good candidates for analysis using derivatives. A key property of polynomials is that they are smooth and continuous, meaning no sharp turns or breaks, and their derivatives exist everywhere on the real line.
Monotonic Function
A monotonic function is one that is entirely non-increasing or non-decreasing. By using the derivative, we can determine the monotonic nature of a function. For \( f(x) = x^3 - 1 \), the derivative \( f'(x) = 3x^2 \) is always non-negative, which means the function is non-decreasing. Since \( f'(x) > 0 \) for all \( x eq 0 \), the function is increasing wherever this condition holds. Thus, \( f(x) \) is increasing everywhere on the real numbers except at \( x = 0 \), a single point where it turns into a constant before increasing again. This understanding allows us to use the function's derivative to conclude its behavior across its domain.

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Most popular questions from this chapter

\(f^{\prime \prime}(x)\) is given. Find \(f(x)\) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if \(f^{\prime \prime}(x)=x\), then \(f^{\prime}(x)=x^{2} / 2+C_{1}\) and \(f(x)=\) \(x^{3} / 6+C_{1} x+C_{2} .\) The constants \(C_{1}\) and \(C_{2}\) cannot be combined because \(C_{1} x\) is not a constant. $$ f^{\prime \prime}(x)=2 \sqrt[3]{x+1} $$

Some software packages can evaluate indefinite integrals. Use your software on each of the following. (a) \(\int 6 \sin (3(x-2)) d x\) (b) \(\int \sin ^{3}(x / 6) d x\) (c) \(\int\left(x^{2} \cos 2 x+x \sin 2 x\right) d x\)

For the price function defined by $$ p(x)=(182-x / 36)^{1 / 2} $$ find the number of units \(x_{1}\) that makes the total revenue a maximum and state the maximum possible revenue. What is the marginal revenue when the optimum number of units, \(x_{1}\), is sold?

According to Torricelli's Law, the time rate of change of the volume \(V\) of water in a draining tank is proportional to the square root of the water's depth. A cylindrical tank of radius \(10 / \sqrt{\pi}\) centimeters and height 16 centimeters, which was full initially, took 40 seconds to drain. (a) Write the differential equation for \(V\) at time \(t\) and the two corresponding conditions. (b) Solve the differential equation. (c) Find the volume of water after 10 seconds.

Brass is produced in long rolls of a thin sheet. To monitor the quality, inspectors select at random a piece of the sheet, measure its area, and count the number of surface imperfections on that piece. The area varies from piece to piece. The following table gives data on the area (in square feet) of the selected piece and the number of surface imperfections found on that piece. $$ \begin{array}{ccc} \hline \text { Piece } & \begin{array}{c} \text { Area in } \\ \text { Square Feet } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Surface Imperfections } \end{array} \\ \hline 1 & 1.0 & 3 \\ 2 & 4.0 & 12 \\ 3 & 3.6 & 9 \\ 4 & 1.5 & 5 \\ 5 & 3.0 & 8 \\ \hline \end{array} $$ (a) Make a scatter plot with area on the horizontal axis and number of surface imperfections on the vertical axis. (b) Does it look like a line through the origin would be a good model for these data? Explain. (c) Find the equation of the least-squares line through the origin. (d) Use the result of part (c) to predict how many surface imperfections there would be on a sheet with area \(2.0\) square feet
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