Problem 25
Question
This problem illustrates the fact that \(f^{\prime \prime}(c)=0\) is not a sufficient condition for an inflection point of a twice-differentiable function.] Show that the function \(f(x)=x^{4}\) has \(f^{\prime \prime}(0)=0\) but that \(f^{\prime \prime}(x)\) does not change sign at \(x=0\) and, hence, \(f(x)\) does not have an inflection point at \(x=0\).
Step-by-Step Solution
Verified Answer
The function \(f(x)=x^4\) does not have an inflection point at \(x=0\) because \(f''(x)\) does not change sign there.
1Step 1: Differentiate the function
Start by finding the first derivative of the function \(f(x) = x^4\). This will involve using the power rule, which states that \(\frac{d}{dx} x^n = nx^{n-1}\). Thus, \(f'(x) = 4x^3\).
2Step 2: Differentiate the first derivative
Next, find the second derivative \(f''(x)\) by differentiating \(f'(x) = 4x^3\) again. Using the power rule gives \(f''(x) = 12x^2\).
3Step 3: Evaluate the second derivative at x=0
Substitute \(x = 0\) into the second derivative \(f''(x) = 12x^2\) to get \(f''(0) = 12(0)^2 = 0\).
4Step 4: Analyze the sign of the second derivative
Look at \(f''(x) = 12x^2\) around \(x=0\). Notice that since \(x^2\) is always non-negative, \(12x^2\) is also always non-negative. Thus, \(f''(x)\) is always \(\geq 0\), implying it does not change sign around \(x=0\).
5Step 5: Conclusion about the inflection point
Since \(f''(x)\) does not change sign around \(x=0\), the point \(x=0\) is not an inflection point for the function \(f(x) = x^4\), despite \(f''(0) = 0\).
Key Concepts
Understanding the Second DerivativeThe Importance of Sign ChangeApplication of the Power Rule
Understanding the Second Derivative
When analyzing functions, the second derivative, denoted as \(f''(x)\), provides valuable insight into the concavity of the function. In simple terms, it shows us how the slope of the original function \(f(x)\) is changing. Whether the function is curving upwards or downwards can be determined by the sign of \(f''(x)\).
For a function to have an inflection point at a particular point \(c\), the second derivative must change sign as we pass through \(c\). This will indicate a change in the direction of the curve of the function from concave up to concave down or vice versa.
In our original exercise, once we calculate \(f''(x)\) for \(f(x) = x^4\), we find that \(f''(x) = 12x^2\). While \(f''(0) = 0\) at \(x = 0\), this alone is not enough to conclude that there is an inflection point. We need to see a sign change of \(f''(x)\) for it to be an inflection point.
For a function to have an inflection point at a particular point \(c\), the second derivative must change sign as we pass through \(c\). This will indicate a change in the direction of the curve of the function from concave up to concave down or vice versa.
In our original exercise, once we calculate \(f''(x)\) for \(f(x) = x^4\), we find that \(f''(x) = 12x^2\). While \(f''(0) = 0\) at \(x = 0\), this alone is not enough to conclude that there is an inflection point. We need to see a sign change of \(f''(x)\) for it to be an inflection point.
The Importance of Sign Change
An inflection point is not just about the second derivative being zero. It's crucial that there is a sign change in \(f''(x)\) around the point in question. Without this sign change, the curve's concavity doesn't switch direction, meaning the behavior of the curve remains consistent.
In the function \(f(x) = x^4\), \(f''(x) = 12x^2\) is derived. Observing \(f''(x)\), we see that, because \(x^2\) is non-negative for all real \(x\), \(f''(x)\) remains non-negative regardless of \(x\). This lack of change in the sign indicates that the curve maintains a consistent concave upward posture near \(x = 0\).
Without this sign change, despite \(f''(0) = 0\), \(x = 0\) cannot be an inflection point for \(f(x) = x^4\). This illustrates that both conditions, a zero second derivative and a sign change, are integral to identifying true inflection points.
In the function \(f(x) = x^4\), \(f''(x) = 12x^2\) is derived. Observing \(f''(x)\), we see that, because \(x^2\) is non-negative for all real \(x\), \(f''(x)\) remains non-negative regardless of \(x\). This lack of change in the sign indicates that the curve maintains a consistent concave upward posture near \(x = 0\).
Without this sign change, despite \(f''(0) = 0\), \(x = 0\) cannot be an inflection point for \(f(x) = x^4\). This illustrates that both conditions, a zero second derivative and a sign change, are integral to identifying true inflection points.
Application of the Power Rule
The power rule is a fundamental tool in calculus for differentiating functions. It provides a quick way to find the derivative of powers of \(x\). The rule states that if you have a term \(x^n\), its derivative is \(nx^{n-1}\).
When given the function \(f(x) = x^4\), applying the power rule allows us to easily find the first derivative \(f'(x) = 4x^3\). Similarly, applying the power rule again to \(f'(x)\) yields the second derivative \(f''(x) = 12x^2\).
Using the power rule simplifies the often complex process of differentiation, making it efficient to find how the function changes, and ultimately aids in analyzing inflection points. Here, recognizing how \(f''(x) = 12x^2\) behaves with respect to \(x\) is critical for judging the function's concavity and whether an inflection point is present.
When given the function \(f(x) = x^4\), applying the power rule allows us to easily find the first derivative \(f'(x) = 4x^3\). Similarly, applying the power rule again to \(f'(x)\) yields the second derivative \(f''(x) = 12x^2\).
Using the power rule simplifies the often complex process of differentiation, making it efficient to find how the function changes, and ultimately aids in analyzing inflection points. Here, recognizing how \(f''(x) = 12x^2\) behaves with respect to \(x\) is critical for judging the function's concavity and whether an inflection point is present.
Other exercises in this chapter
Problem 24
Find \(c\) such that \(f^{\prime}(c)=0\) and determine whether \(f(x)\) has a local extremum at \(x=c .\) \(f(x)=x^{5}\)
View solution Problem 25
Use l'Hospital's rule to find the limits. $$ \lim _{x \rightarrow \infty} x e^{-x} $$
View solution Problem 25
Consider density-dependent population growth models of the form $$ N_{t+1}=R\left(N_{t}\right) N_{t} $$ The function \(R(N)\) describes the per capita growth. V
View solution Problem 25
Suppose that \(f\) is twice differentiable on an open interval \(I\). Show that if \(f^{\prime \prime}(x)
View solution