Problem 50
Question
Find the inflection point(s), if any, of each function. $$ g(x)=x^{3}-6 x $$
Step-by-Step Solution
Verified Answer
The inflection point of the function \(g(x) = x^3 - 6x\) is at coordinate (0, 0).
1Step 1: Find the second derivative of the given function.
Given function:
\(
g(x)=x^{3}-6x
\)
First, find the first derivative:
\(
g'(x) = \frac{d}{dx}(x^3 - 6x)
\)
Using the power rule, we get:
\(
g'(x) = 3x^2 - 6
\)
Next, find the second derivative:
\(
g''(x) = \frac{d}{dx}(3x^2 - 6)
\)
Using the power rule, we get:
\(
g''(x) = 6x
\)
2Step 2: Solve for x when the second derivative is equal to 0.
Now, set the second derivative equal to 0 and solve for x:
\(
6x = 0
\)
Divide both sides by 6:
\(
x = 0
\)
3Step 3: Test for change in concavity on either side of the critical point.
We need to check if there is a change in concavity on either side of the critical point \(x = 0\).
Testing the second derivative at a point to the left of 0, say x = -1:
\(
g''(-1) = 6(-1) = -6
\)
which indicates that the graph is concave down in this region.
Testing the second derivative at a point to the right of 0, say x = 1:
\(
g''(1) = 6(1) = 6
\)
which indicates that the graph is concave up in this region.
Since there is a change in concavity at \(x = 0\), that means we found an inflection point.
To find the coordinates of the inflection point, substitute \(x = 0\) into the original function:
\(
g(0) = 0^3 - 6(0) = 0
\)
So, the inflection point is at coordinate (0, 0).
Key Concepts
ConcavitySecond Derivative TestCritical Points
Concavity
Understanding concavity is crucial for analyzing the shape of graphs of functions. Concavity refers to the curvature's direction in a particular section of a graph. Essentially, a graph can exhibit two types of concavity: concave up or concave down.
When a function is concave up, the graph curves upward, resembling the shape of a cup that could hold water. You can visualize this by imagining any segment of the graph is bendable and could form part of a smile. In contrast, a function is concave down when the graph bends downward like a frown and would cause water to spill. Mathematically, concavity is determined by the second derivative of a function.
For a function with a continuous second derivative, if the second derivative is positive over an interval, the function is concave up on that interval. Conversely, if the second derivative is negative over an interval, the function is concave down. A change in concavity, where a graph transitions from concave up to concave down or vice versa, is indicative of a particular point called an inflection point.
When a function is concave up, the graph curves upward, resembling the shape of a cup that could hold water. You can visualize this by imagining any segment of the graph is bendable and could form part of a smile. In contrast, a function is concave down when the graph bends downward like a frown and would cause water to spill. Mathematically, concavity is determined by the second derivative of a function.
For a function with a continuous second derivative, if the second derivative is positive over an interval, the function is concave up on that interval. Conversely, if the second derivative is negative over an interval, the function is concave down. A change in concavity, where a graph transitions from concave up to concave down or vice versa, is indicative of a particular point called an inflection point.
Second Derivative Test
The second derivative test is a powerful tool for identifying the concavity of a function and locating its inflection points. The test involves taking the second derivative of the function and determining its sign. If the second derivative of a function is greater than zero, \( g''(x) > 0 \), the function is concave up. Conversely, if it is less than zero, \( g''(x) < 0 \), the function is concave down.
The point where the second derivative equals zero is particularly interesting because it could be an inflection point, where the concavity changes. However, to confirm it as an inflection point, we must show that the second derivative changes sign around the critical point. If the second derivative does not change sign—meaning it's positive (or negative) on both sides of the critical point—that critical point is not an inflection point.
The point where the second derivative equals zero is particularly interesting because it could be an inflection point, where the concavity changes. However, to confirm it as an inflection point, we must show that the second derivative changes sign around the critical point. If the second derivative does not change sign—meaning it's positive (or negative) on both sides of the critical point—that critical point is not an inflection point.
Critical Points
Critical points are essential locations on a function's graph where the derivative is either zero or undefined. These points are central to analyzing a function since they may correspond to local maxima, minima, or points of inflection.
To find the critical points of a function, we first determine its first derivative and then solve for the values of \( x \) where this derivative is zero or does not exist. For the given function \( g(x) = x^3 - 6x \), the first derivative is \( g'(x) = 3x^2 - 6 \), and its critical points are found by setting \( g'(x) = 0 \) and solving for \( x \).
However, not all critical points correspond to inflection points. To identify an inflection point, we use the second derivative test to check if there is a change in the concavity of the function at the critical point. In our exercise, \( x = 0 \) was found to be a critical point and ultimately an inflection point because there is a change in concavity at that point, as evidenced by the second derivative changing from negative to positive.
To find the critical points of a function, we first determine its first derivative and then solve for the values of \( x \) where this derivative is zero or does not exist. For the given function \( g(x) = x^3 - 6x \), the first derivative is \( g'(x) = 3x^2 - 6 \), and its critical points are found by setting \( g'(x) = 0 \) and solving for \( x \).
However, not all critical points correspond to inflection points. To identify an inflection point, we use the second derivative test to check if there is a change in the concavity of the function at the critical point. In our exercise, \( x = 0 \) was found to be a critical point and ultimately an inflection point because there is a change in concavity at that point, as evidenced by the second derivative changing from negative to positive.
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