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Chapter 3: Problem 42

For each function, find all critical numbers and then use the second- derivative test to determine whether the function has a relative maximum or minimum at each critical number. $$ f(x)=x^{3}-12 x+4 $$

Short Answer

Expert verified
Critical numbers are \( x = 2 \) (minimum) and \( x = -2 \) (maximum).

Step by step solution

01

Find the First Derivative

To find the critical numbers for the function \( f(x) = x^3 - 12x + 4 \), we first need to find its first derivative. The first derivative \( f'(x) \) is computed as follows:\[f'(x) = 3x^2 - 12\]
02

Set the First Derivative to Zero

Next, we set the first derivative equal to zero to find the critical numbers of the function:\[3x^2 - 12 = 0\]Simplify this equation by dividing all terms by 3:\[x^2 - 4 = 0\]
03

Solve for Critical Numbers

Solve the simplified equation from Step 2:\[x^2 - 4 = 0\]Factor the equation:\[(x - 2)(x + 2) = 0\]Thus, the critical numbers are \( x = 2 \) and \( x = -2 \).
04

Find the Second Derivative

To apply the second-derivative test, we first find the second derivative of \( f(x) \). Start by differentiating the first derivative \( f'(x) = 3x^2 - 12 \):\[f''(x) = 6x\]
05

Apply the Second-Derivative Test at Critical Numbers

Evaluate the second derivative at each critical number:1. For \( x = 2 \): \[f''(2) = 6(2) = 12\] Since \( f''(2) > 0 \), \( f(x) \) has a relative minimum at \( x = 2 \).2. For \( x = -2 \): \[f''(-2) = 6(-2) = -12\] Since \( f''(-2) < 0 \), \( f(x) \) has a relative maximum at \( x = -2 \).

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Key Concepts

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

First Derivative
The first step in finding critical numbers for a function is to determine the first derivative. The first derivative of a function, denoted as \( f'(x) \), represents the rate at which the function's value changes at any given point. It's essentially a formula for the slope of the tangent line at every point on the curve of the function. To find this, differentiate the original function with respect to \( x \). For \( f(x) = x^3 - 12x + 4 \), the first derivative is calculated as:
  • Differentiate \( x^3 \) to get \( 3x^2 \).
  • Differentiate \(-12x \) to get \(-12 \).
  • As constants disappear when differentiated, \( +4 \) becomes \( 0 \).
Therefore, the first derivative is \( f'(x) = 3x^2 - 12 \).
Finding the critical numbers involves setting this first derivative equal to zero, because critical points occur where the rate of change (slope) shifts direction.
Second Derivative
The second derivative of a function, denoted as \( f''(x) \), helps us understand the concavity of the function's graph. Concavity tells us whether the graph bends upwards or downwards at certain points, providing deeper insights into the behavior of the function around its critical numbers.
To find the second derivative, differentiate the first derivative. For the function \( f(x) \) with \( f'(x) = 3x^2 - 12 \), differentiating again gives:
  • Differentiate \( 3x^2 \) to get \( 6x \).
  • Differentiate \(-12 \) to get \( 0 \), as it is a constant.
Thus, the second derivative is \( f''(x) = 6x \). This second derivative will be pivotal in the next analysis step鈥攖he second-derivative test, revealing more about the nature of critical points.
Second-Derivative Test
The second-derivative test is a method used to evaluate whether a critical number is a point of local maxima, minima, or a saddle point. Here's how it works:
  • If \( f''(x) > 0 \) at a critical number, the function is concave up, suggesting a local minimum at that point.
  • If \( f''(x) < 0 \) at a critical number, the function is concave down, indicating a local maximum.
  • If \( f''(x) = 0 \), the test is inconclusive at that point.
For our function \( f(x) = x^3 - 12x + 4 \), applying the test gives:- At \( x = 2 \), \( f''(2) = 12 \), which is greater than 0, indicating a local minimum.
- At \( x = -2 \), \( f''(-2) = -12 \), which is less than 0, signaling a local maximum.
This makes the second-derivative test a powerful tool to classify critical points effectively.

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Most popular questions from this chapter

Suppose that you have a formula that relates the amount of gas used (denoted by \(x\) ) to the distance driven (denoted by \(y\) ) in your car. State, in everyday language, what \(\frac{d y}{d x}\) and \(\frac{d x}{d y}\) would mean.

If an epidemic spreads through a town at a rate that is proportional to the number of infected people and to the number of uninfected people, then the rate is \(R(x)=c x(p-x)\), where \(x\) is the number of infected people and \(c\) and \(p\) (the population) are positive constants. Show that the rate \(R(x)\) is greatest when half of the population is infected.

An electronics store can sell 35 cellular telephones per week at a price of \(\$ 200\). The manager estimates that for each \(\$ 20\) price reduction she can sell 9 more per week. The telephones cost the store \(\$ 100\) each, and fixed costs are \(\$ 700\) per week. a. If \(x\) is the number of \(\$ 20\) price reductions, find the price \(p(x)\) and enter it in \(y_{1}\). Then enter the quantity function \(q(x)\) in \(y_{2}\). b. Make \(y_{3}\) the revenue function by defining \(y_{3}=y_{1} y_{2} \quad\) (price times quantity). c. Make \(y_{4}\) the cost function by defining \(y_{4}\) as unit cost times \(y_{2}\) plus fixed costs. d. Make \(y_{5}\) the profit function by defining \(y_{5}=y_{3}-y_{4} \quad\) (revenue minus cost). e. Turn off \(y_{1}, y_{2}, y_{3}\), and \(y_{4}\) and graph the profit function \(y_{5}\) for \(x\) -values \([-10,10]\), using TABLE or TRACE to find an appropriate \(y\) -interval. Then use MAXIMUM to maximize it. f. Use EVALUATE to find the price and the quantity for this maximum profit.

If an epidemic spreads through a town at a rate that is proportional to the number of uninfected people and to the square of the number of infected people, then the rate is \(R(x)=c x^{2}(p-x)\), where \(x\) is the number of infected people and \(c\) and \(p\) (the population) are positive constants. Show that the rate \(R(x)\) is greatest when two-thirds of the population is infected.

True or False: If \(f\) has an absolute maximum value, then \(-f\) will have an absolute minimum value.
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