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Chapter 2: Problem 14

Sketch the graphs of the following functions. $$f(x)=2 x^{3}+x-2$$

Short Answer

Expert verified
Identify the function type, find derivatives, identify inflection points, calculate key points, and plot.

Step by step solution

01

Identify the type of function

The function provided is a cubic polynomial, as it is in the form of a polynomial with a degree of 3. Specifically, it is given by the equation: \[ f(x) = 2x^3 + x - 2 \]
02

Find the critical points

Standard practice is to find the first derivative and set it to zero to identify critical points. Compute the first derivative: \[ f'(x) = \frac{d}{dx} (2x^3 + x - 2) = 6x^2 + 1 \] Set the first derivative equal to zero: \[ 6x^2 + 1 = 0 \] Solve for x: \[ 6x^2 = -1 \] No real roots exist for this equation since the discriminant is negative. Thus, there are no critical points.
03

Determine the concavity and inflection points

Find the second derivative to determine concavity and locate any inflection points: \[ f''(x) = \frac{d}{dx} (6x^2 + 1) = 12x \] Set the second derivative equal to zero to find inflection points: \[ 12x = 0 \] Solve for x: \[ x = 0 \]The function has an inflection point at \(x = 0\). Evaluate the function at this point: \[ f(0) = 2(0)^3 + 0 - 2 = -2 \] The inflection point is at (0, -2).
04

Evaluate the function at key points

Evaluate f(x) at several key x-values to gain insight: \[ f(-2) = 2(-2)^3 + (-2) - 2 = -16 - 2 - 2 = -20 \] \[ f(-1) = 2(-1)^3 + (-1) - 2 = -2 - 1 - 2 = -5 \] \[ f(1) = 2(1)^3 + 1 - 2 = 2 + 1 - 2 = 1 \] \[ f(2) = 2(2)^3 + 2 - 2 = 16 + 2 - 2 = 16 \]
05

Plot the function

Using the evaluated points and the inflection point, sketch the graph. Plot points such as (-2, -20), (-1, -5), (0, -2), (1, 1), and (2, 16) and then draw a smooth curve connecting these points. Ensure the graph passes through these points and shows the correct concavity.

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Key Concepts

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

cubic functions
A cubic function is a polynomial function of degree three. This means the highest exponent of the variable is three. Generally, a cubic polynomial looks like \( f(x) = ax^3 + bx^2 + cx + d \). In the given exercise, the function is \( f(x) = 2x^3 + x - 2 \). Cubic functions can have interesting characteristics:
  • They can have one or two bends.

  • They always cross the x-axis at least once because they have at least one real root.

  • They can have up to three real roots.

  • Their graphs can show various symmetries.

The goal when sketching cubic functions is to understand its overall shape, which often involves looking at its critical points, concavity, and inflection points.
critical points
Critical points are points on the graph where the function's slope is zero or undefined. To find these points, we compute the first derivative and set it to zero. For our function \( f(x) = 2x^3 + x - 2 \), the first derivative is \( f'(x) = 6x^2 + 1 \). Setting this equal to zero and solving: \( 6x^2 + 1 = 0 \) leads to \(x \), we get
  • 6x虏 = -1.

Since the right side is negative, there are no real solutions. Hence, there are no real critical points for this function.
first and second derivatives
The first derivative of a function provides us with the slope of the original function. It helps us find critical points where these slopes (or tangents) are horizontal, i.e., where the slope is zero. For the function \( f(x) = 2x^3 + x - 2 \), the first derivative is \( f'(x) = 6x^2 + 1 \).To understand concavity and find inflection points, we use the second derivative, which is the derivative of the first derivative. The second derivative for \( f(x) \) is \( f''(x) = 12x \). Inflection points occur where \( f''(x) = 0 \): \( 12x = 0 \), which means
  • x = 0.

This does not tell us where the function changes concavity.
inflection points
Inflection points are where the graph changes concavity, going from concave up to concave down or vice versa. This point is found by setting the second derivative to zero: For \( f(x) = 2x^3 + x - 2 \), \( f''(x) = 12x \) setting this to zero yields one solution:
  • x = 0.

We substitute this x-value back into the original function to find the corresponding y-value: \(f(0) = -2 \). Therefore, the inflection point is at (0, -2). This means the graph changes its shape at this point.

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Most popular questions from this chapter

Each of the graphs of the functions has one relative extreme point. Plot this point and check the concavity there. Using only this information, sketch the graph. [Recall that if \(f(x)=a x^{2}+b x+c,\) then \(f(x)\) has a relative minimum point when \(a>0\) and a relative maximum point when \(a<0.1\) $$f(x)=-x^{2}-8 x-10$$

A one-product firm estimates that its daily total cost function (in suitable units) is \(C(x)=x^{3}-6 x^{2}+13 x+15\) and its total revenue function is \(R(x)=28 x .\) Find the value of \(x\) that maximizes the daily profit.

The revenue function for a one product firm is $$R(x)=200-\frac{1600}{x+8}-x.$$ Find the value of \(x\) that results in maximum revenue.

Coffee consumption in the United States is greater on a per capita basis than anywhere else in the world. However, due to price fluctuations of coffee beans and worries over the health effects of caffeine, coffee consumption has varied considerably over the years. According to data published in The Wall Street Journal, the number of cups \(f(x)\) consumed daily per adult in year \(x\) (with 1955 corresponding to \(x=0\) ) is given by the mathematical model $$f(x)=2.77+0.0848 x-0.00832 x^{2}+0.000144 x^{3}.$$ (a) Graph \(y=f(x)\) to show daily coffee consumption from 1955 through 1994. (b) Use \(f^{\prime}(x)\) to determine the year in which coffee consumption was least during this period. What was the daily coffee consumption at that time? (c) Use \(f^{\prime}(x)\) to determine the year in which coffee consumption was greatest during this period. What was the daily coffee consumption at that time? (d) Use \(f^{\prime \prime}(x)\) to determine the year in which coffee consumption was decreasing at the greatest rate.

Some years ago, it was estimated that the demand for steel approximately satisfied the equation \(p=256-50 x\) and the total cost of producing \(x\) units of steel was \(a(x)=182+56 x\). (The quantity \(x\) was measured in millions of tons and the price and total cost were measured in millions of dollars.) Determine the level of production and the corresponding price that maximize the profits.
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