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Chapter 4: Problem 19

Write the first and second derivatives of the function and use the second derivative to determine inputs at which inflection points might exist. \(f(x)=x^{3}-6 x^{2}+12 x\)

Short Answer

Expert verified
The function has an inflection point at \( x = 2 \).

Step by step solution

01

Find the First Derivative

To find the first derivative of the function \( f(x) = x^3 - 6x^2 + 12x \), we differentiate each term with respect to \( x \). \( f'(x) = \frac{d}{dx}(x^3) - \frac{d}{dx}(6x^2) + \frac{d}{dx}(12x) \) \( f'(x) = 3x^2 - 12x + 12 \).
02

Find the Second Derivative

Next, we find the second derivative by differentiating the first derivative. \( f''(x) = \frac{d}{dx}(3x^2 - 12x + 12) \) \( f''(x) = 6x - 12 \).
03

Determine Potential Inflection Points

To find potential inflection points, set the second derivative equal to zero and solve for \( x \). \( 6x - 12 = 0 \) Solve for \( x \):\( 6x = 12 \)\( x = 2 \).
04

Verify Change in Concavity

Check the sign of the second derivative before and after \( x = 2 \) to confirm an inflection point. Choose values: \( x = 1 \) and \( x = 3 \).At \( x = 1 \), \( f''(1) = 6(1) - 12 = -6 \) (negative).At \( x = 3 \), \( f''(3) = 6(3) - 12 = 6 \) (positive).Since \( f''(x) \) changes from negative to positive around \( x = 2 \), it confirms an inflection point 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 derivative of a function gives us insight into the rate at which the function's value is changing. For the function \( f(x) = x^3 - 6x^2 + 12x \), the first derivative is found by differentiating each term separately. This results in the derivative \( f'(x) = 3x^2 - 12x + 12 \).
This expression represents the slope of the tangent line to the curve at any point \( x \). If \( f'(x) > 0 \), the function is increasing at that point. If \( f'(x) < 0 \), the function is decreasing.
The first derivative is crucial for identifying intervals of increase or decrease and is the foundation for further analysis using the second derivative.
Second Derivative
The second derivative is the derivative of the first derivative and provides information about the concavity of the function. In our example, the first derivative \( f'(x) = 3x^2 - 12x + 12 \) leads to a second derivative \( f''(x) = 6x - 12 \).
This second derivative tells us how the rate of change of the function's slope is varying.
  • If \( f''(x) > 0 \), the function is concave up, resembling a smile.
  • If \( f''(x) < 0 \), the function is concave down, resembling a frown.
These insights into concavity help identify possible inflection points and provide a deeper understanding of the function's behavior.
Inflection Points
Inflection points are points on the graph where the concavity changes. For our function, we find these by setting the second derivative \( f''(x) = 6x - 12 \) equal to zero. Solving \( 6x - 12 = 0 \) gives \( x = 2 \).
Inflection points are special because, at these points, the curve of the function transitions from being concave up to concave down, or vice versa. To confirm that \( x = 2 \) is indeed an inflection point, we further verify by looking at the sign of the second derivative around this value.
This ensures that a change in concavity actually occurs.
Change in Concavity
A change in concavity is confirmed when the sign of the second derivative switches from positive to negative or from negative to positive. By checking \( f''(1) = -6 \) and \( f''(3) = 6 \), we see a change in sign around \( x = 2 \). This change signifies an inflection point.
Understanding changes in concavity gives a complete picture of how the function behaves. This information, combined with insights from both derivatives, can help predict and analyze the function's overall shape.
Recognizing these changes is essential in calculus for graphing functions and for applications across physics, economics, and engineering, where understanding how a system progresses is vital.

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Most popular questions from this chapter

Sketch the graph of a function \(f\) such that all of the following statements are true. \- \(f\) has a relative minimum at \(x=3\). \- \(f\) has a relative maximum at \(x=-1\). \- \(f^{\prime}(x)>0\) for \(x<-1\) and \(x>3\) \- \(f^{\prime}(x)<0\) for \(-1

The body-mass index (BMI) of an individual who weighs \(w\) pounds and is \(h\) inches tall is given as $$ B=703 \frac{w}{h^{2}} $$ (Source: \(C D C)\) a. Write an equation showing the relationship between the body-mass index and weight of a woman who is 5 feet 8 inches tall. b. Construct a related-rates equation showing the interconnection between the rates of change with respect to time of the weight and the body-mass index. c. Consider a woman who weighs 160 pounds and is 5 feet 8 inches tall. If \(\frac{d w}{d t}=1\) pound per month, evaluate and interpret \(\frac{d B}{d t}\). d. Suppose a woman who is 5 feet 8 inches tall has a bodymass index of 24 points. If her body-mass index is decreasing by 0.1 point per month, at what rate is her weight changing?

Sketch the graph of a function \(f\) such that all of the following statements are true. \- \(f^{\prime}(x)>0\) for \(x<-1\) \- \(f^{\prime}(x)<0\) for \(x>-1\) \- \(f^{\prime}(-1)=0\)

a. Write the derivative formula. b. Locate any relative extreme points and identify the extreme as a maximum or minimum. $$ h(x)=x^{3}-8 x^{2}-6 x $$

If they exist, find the absolute extremes of \(y=\frac{2-3 x+x^{2}}{(3.5+x)^{2}}\) over all real number inputs. If an absolute maximum or absolute minimum does not exist, explain why.
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