Mistakes when finding inflection points: second derivative undefined | AP Calculus AB | Khan Academy

Khan Academy
7 Sept 201706:11
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TLDRIn this instructional video, the focus is on identifying inflection points of the function g(x) = x^(1/3). The instructor first reviews Robert's solution, which initially appears correct in finding the first and second derivatives. However, Robert incorrectly concludes that there are no inflection points, missing the fact that an inflection point occurs where the second derivative is undefined, specifically at x=0. The instructor corrects this by demonstrating that the concavity of the function changes sign as x crosses zero, confirming an inflection point at x=0. The video serves as a lesson on the importance of considering undefined points in addition to where derivatives equal zero when searching for inflection points.

Takeaways
  • 📚 The original function g(x) is the cube root of x, which can be expressed as x^(1/3).
  • 🔑 Robert correctly found the first derivative using the power rule, resulting in (1/3)x^(-2/3).
  • 📉 For the second derivative, Robert applied the power rule again, correctly obtaining -(2/9)x^(-5/3).
  • 🧐 Robert recognized that the second derivative can be simplified to -2/9 divided by the cube root of x^5, which is equivalent to x^(5/3).
  • 🔍 Robert attempted to find x values where the second derivative equals zero but correctly concluded that there are no such solutions.
  • 🤔 Robert incorrectly stated that g(x) does not have any inflection points, missing the case where the second derivative could be undefined.
  • ❗ The second derivative g''(x) is undefined at x = 0, which should be considered as a candidate for an inflection point.
  • 📋 To determine if x = 0 is an inflection point, one must check if the second derivative changes signs around this value.
  • 📊 By testing values less than and greater than zero, it's shown that the concavity of g(x) does indeed change sign at x = 0.
  • 🔄 At x < 0, the second derivative is positive, indicating concave up behavior, while at x > 0, it's negative, indicating concave down behavior.
  • ✅ The correct conclusion is that there is an inflection point at x = 0, where the second derivative is undefined, not where it equals zero.
Q & A

  • What is the function g(x) that Robert is analyzing?

    -The function g(x) that Robert is analyzing is the cube root of x, which can also be expressed as x to the power of 1/3.

  • What is the first step in finding inflection points of a function?

    -The first step in finding inflection points of a function is to find the first and second derivatives of the function.

  • How does Robert find the first derivative of g(x)?

    -Robert uses the power rule to find the first derivative of g(x), which results in 1/3x to the power of (1/3 - 1) = 1/3x to the power of -2/3.

  • What is the second derivative of g(x) according to Robert's solution?

    -According to Robert's solution, the second derivative of g(x) is -2/9x to the power of -5/3.

  • Why does Robert's solution initially seem correct for the second derivative?

    -Robert's solution initially seems correct because he correctly applies the power rule and simplifies the expression for the second derivative.

  • What is the mistake Robert makes in his third step?

    -Robert's mistake in the third step is concluding that there are no inflection points for g(x) without considering the possibility of the second derivative being undefined.

  • Why can't the second derivative of g(x) be equal to zero?

    -The second derivative of g(x) can't be equal to zero because the numerator (2) is a constant and cannot be zero, making it impossible for the entire expression to equal zero.

  • Where is the second derivative of g(x) undefined?

    -The second derivative of g(x) is undefined at x = 0, because you would be dividing by zero in the expression for the second derivative.

  • What does it mean for a function to have an inflection point where the second derivative is undefined?

    -For a function to have an inflection point where the second derivative is undefined means that there is a point on the graph where the concavity changes, but this change cannot be detected by setting the second derivative equal to zero.

  • How can you determine if x = 0 is an inflection point for g(x)?

    -You can determine if x = 0 is an inflection point for g(x) by testing the sign of the second derivative on either side of x = 0 and observing if there is a change in concavity.

  • What is the concavity of g(x) for values less than zero?

    -For values less than zero, the concavity of g(x) is upwards because the second derivative is positive when x is negative.

  • What is the concavity of g(x) for values greater than zero?

    -For values greater than zero, the concavity of g(x) is downwards because the second derivative is negative when x is positive.

  • How does the concavity of g(x) change at x = 0?

    -The concavity of g(x) changes at x = 0 from being upwards when x is less than zero to being downwards when x is greater than zero.

  • What is the correct conclusion about the inflection point of g(x)?

    -The correct conclusion is that there is an inflection point at x = 0, where the second derivative is undefined and the concavity changes.

Outlines
00:00
📚 Analysis of Robert's Solution for Inflection Points

This paragraph discusses the analysis of Robert's solution to find inflection points of the function g(x) = x^(1/3). The instructor begins by explaining the process of finding the first and second derivatives using the power rule. The first derivative is calculated as 1/3x^(-2/3), and the second derivative is found to be -2/9x^(-5/3). The instructor then points out that Robert correctly rewrote the second derivative as -2/9(x^5)/(x^3). Robert's next step was to find where the second derivative equals zero, concluding correctly that there is no such x value that would make the second derivative zero. However, the instructor notes that Robert's conclusion that g has no inflection points is premature. Inflection points can also occur where the second derivative is undefined, such as at x = 0, which Robert failed to consider. The instructor then explains how to test for inflection points by examining the sign of the second derivative across different intervals, which Robert did not do.

05:01
📉 Correction of Robert's Conclusion on Inflection Points

In this paragraph, the instructor corrects Robert's conclusion by demonstrating that the function g(x) does indeed have an inflection point. The instructor points out that while Robert was correct in stating that the second derivative does not equal zero, he overlooked the fact that an inflection point can occur where the second derivative is undefined. The instructor shows that the second derivative is undefined at x = 0, which is a candidate for an inflection point. To confirm this, the instructor sets up a table to test the concavity of the function across intervals around x = 0. Using test values of x = -1 and x = 1, the instructor determines that the concavity of the function changes sign at x = 0, indicating that there is indeed an inflection point at this point. The instructor concludes that Robert's error was in not considering the undefined second derivative as a potential source of inflection points.

Mindmap
Keywords
💡Inflection Point
An inflection point is a point on a curve where the concavity changes. In the context of the video, the instructor is discussing the inflection point of the function g(x) = x^(1/3), which is the cube root of x. The video aims to find where this function changes concavity, which is a critical concept in calculus for understanding the shape of curves. The script mentions that Robert incorrectly concluded there were no inflection points, but the instructor corrects this by showing that there is an inflection point at x = 0.
💡Cube Root
The cube root of a number x, denoted as x^(1/3), is a value that, when multiplied by itself three times, gives the original number x. In the video, the function g(x) is defined as the cube root of x, and the script uses this function to explore the concept of inflection points. The cube root is fundamental to the function being analyzed and is the starting point for finding its derivatives.
💡Derivative
In calculus, a derivative represents the rate at which a function changes with respect to one of its variables. The first derivative of a function gives the slope of the tangent line to the function at any point, while the second derivative indicates the concavity of the function. The video script discusses finding the first and second derivatives of g(x) to determine the inflection points.
💡Power Rule
The power rule is a basic principle in calculus for finding the derivative of a function that is a power of the variable. It states that the derivative of x^n (where n is a constant) is nx^(n-1). In the script, the power rule is used to find the first derivative of g(x) = x^(1/3), resulting in g'(x) = (1/3)x^(-2/3).
💡Second Derivative
The second derivative of a function is the derivative of the first derivative. It provides information about the function's concavity and is used to identify inflection points. In the video, the second derivative of g(x) is calculated as g''(x) = -2/9 * x^(-5/3), which is then analyzed to determine the function's concavity and potential inflection points.
💡Concavity
Concavity refers to the curvature of a function. A function is said to be concave up if its graph curves upward like a U, and concave down if it curves downward. The sign of the second derivative is used to determine concavity: if positive, the function is concave up, and if negative, it is concave down. The video discusses how the concavity of g(x) changes at x = 0, indicating an inflection point.
💡Undefined
In mathematics, a function is said to be undefined at a certain point if it does not have a value at that point. In the script, the second derivative g''(x) is found to be undefined at x = 0 because it involves division by zero (x^(5/3)/x^(5/3) = 1/x^(5/3) when x ≠ 0, but 1/0 is undefined). This is a critical observation that leads to the identification of an inflection point.
💡Sign Change
A sign change occurs when the value of a mathematical expression changes from positive to negative or vice versa. In the context of the video, the instructor is looking for a sign change in the second derivative to identify an inflection point. The script demonstrates that the second derivative changes sign as x crosses zero, indicating a change in concavity.
💡Test Values
Test values are specific points chosen to evaluate the behavior of a function or its derivatives. They are used to determine the sign of the derivative over different intervals and to check for sign changes that might indicate an inflection point. In the script, the instructor uses test values of x = -1 and x = 1 to analyze the sign of the second derivative and confirm the inflection point at x = 0.
💡Candidate Inflection Point
A candidate inflection point is a point where the second derivative is zero or undefined, and it could potentially be an inflection point if it satisfies the conditions of changing concavity. In the video, the instructor initially dismisses the possibility of inflection points where the second derivative is zero but later identifies x = 0 as a candidate inflection point because the second derivative is undefined there.
Highlights

Robert is tasked with finding inflection points for the function g(x) = cube root of x.

The original function g(x) is expressed as x to the power of 1/3.

First derivative is calculated using the power rule, resulting in 1/3x to the power of -2/3.

Second derivative involves multiplying the first derivative by 1/3 and adjusting the exponent.

Second derivative is simplified to negative 2/9 times x to the power of -5/3.

Robert's work on finding the first and second derivatives is confirmed as correct.

The second derivative has no solution for x values where it equals zero.

Inflection points are typically where the second derivative is zero or undefined.

Robert incorrectly states that g has no inflection points.

Inflection points can occur where the second derivative is undefined, such as at x = 0.

The second derivative g'' is undefined at x = 0 due to division by zero.

A table is set up to test the concavity and sign of the second derivative across intervals.

Test values are chosen to ensure they are indicative and close to the candidate inflection point.

The concavity of g switches from upwards to downwards as x crosses zero.

The second derivative g'' switches signs as x crosses zero, indicating an inflection point.

The function g is defined at x = 0, confirming an inflection point at this point.

The correct inflection point is identified at x = 0, where the second derivative is undefined.

Robert's mistake in step three is pointed out, as there is indeed an inflection point.

Transcripts

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