Problem 42
Question
Decide whether the statement is true or false. Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=2 x-y .\) Justify your answer. All the inflection points of \(f\) lie on the line \(y=2 x-2\)
Step-by-Step Solution
Verified Answer
True, all inflection points lie on the line \( y = 2x - 2 \).
1Step 1: Find Second Derivative
To determine the points of inflection, we need to find the second derivative of the function. We begin with the given differential equation \( \frac{dy}{dx} = 2x - y \). We need to find \( \frac{d^2y}{dx^2} \). Differentiating both sides with respect to \( x \), we get \( \frac{d}{dx}(\frac{dy}{dx}) = \frac{d}{dx}(2x - y) \). This gives us \( \frac{d^2y}{dx^2} = 2 - \frac{dy}{dx} \). Since \( \frac{dy}{dx} = 2x - y \), substitute back to get \( \frac{d^2y}{dx^2} = 2 - (2x - y) = y - 2x + 2 \).
2Step 2: Identify Inflection Points Condition
An inflection point occurs where the second derivative changes sign, which corresponds to \( \frac{d^2y}{dx^2} = 0 \). Use the expression we found: \( y - 2x + 2 = 0 \). Simplifying, we get \( y = 2x - 2 \). This equation represents the line on which the inflection points of \( f \) must lie.
3Step 3: Compare to the Given Statement
The problem statement claims that all inflection points of \( f \) lie on the line \( y = 2x - 2 \). From Step 2, we've also derived that for the second derivative to be zero, the condition is \( y = 2x - 2 \). This matches the given line in the problem statement.
Key Concepts
Second DerivativeInflection PointsSign Change
Second Derivative
The second derivative is a powerful tool in calculus, often used to gain a deeper understanding of the behaviour of a function. Here, we deal with the second derivative in the context of differential equations and inflection points.
The given differential equation is \( \frac{dy}{dx} = 2x - y \). This equation describes the rate of change of \( y \) with respect to \( x \).
To find the second derivative, \( \frac{d^2y}{dx^2} \), we differentiate this equation once more with respect to \( x \).
Here’s the step-by-step process:
Thus, the second derivative tells us how the rate of change of \( y \) is changing—a concept critical for determining the concavity of a function and finding inflection points.
The given differential equation is \( \frac{dy}{dx} = 2x - y \). This equation describes the rate of change of \( y \) with respect to \( x \).
To find the second derivative, \( \frac{d^2y}{dx^2} \), we differentiate this equation once more with respect to \( x \).
Here’s the step-by-step process:
- Differentiating both sides: \( \frac{d}{dx}(\frac{dy}{dx}) = \frac{d}{dx}(2x - y) \)
- The second derivative simplifies to: \( \frac{d^2y}{dx^2} = y - 2x + 2 \)
Thus, the second derivative tells us how the rate of change of \( y \) is changing—a concept critical for determining the concavity of a function and finding inflection points.
Inflection Points
Inflection points are crucial when studying the graph of a function. They signal where the function's concavity changes, i.e., where it shifts from being concave upwards to concave downwards or vice versa.
To find inflection points, we focus on the second derivative \( \frac{d^2y}{dx^2} \).
An inflection point occurs when:
An important takeaway is that all inflection points of our function \( f \) should fall along this line, satisfying the equation derived from the second derivative.
To find inflection points, we focus on the second derivative \( \frac{d^2y}{dx^2} \).
An inflection point occurs when:
- The second derivative is zero (\( \frac{d^2y}{dx^2} = 0 \))
- The sign of \( \frac{d^2y}{dx^2} \) changes around a particular value of \( x \)
An important takeaway is that all inflection points of our function \( f \) should fall along this line, satisfying the equation derived from the second derivative.
Sign Change
Understanding sign changes in the context of the second derivative is key to identifying inflection points.
For an inflection point to be confirmed,
Without sign changes through this line, inflection points cannot truly manifest as per their definition.
Thus, identifying where the sign change occurs in relation to \( \frac{d^2y}{dx^2} \) is integral in confirming true inflection points on the curve.
For an inflection point to be confirmed,
- The second derivative, \( \frac{d^2y}{dx^2} \), must change its sign around the point. That means it transitions from negative to positive or positive to negative.
- This change in sign indicates a change in concavity, which defines an inflection point.
Without sign changes through this line, inflection points cannot truly manifest as per their definition.
Thus, identifying where the sign change occurs in relation to \( \frac{d^2y}{dx^2} \) is integral in confirming true inflection points on the curve.
Other exercises in this chapter
Problem 41
Solve the differential equations in Problems \(34-43 .\) Assume \(a, b,\) and \(k\) are nonzero constants. $$\frac{d y}{d t}=k y^{2}\left(1+t^{2}\right)$$
View solution Problem 42
Explain what is wrong with the statement. The differential equation \(d P / d t=0.08 P-0.0032 P^{2}\) has one equilibrium solution, at \(P=25\)
View solution Problem 42
Is the statement true or false? Assume that \(y=f(x)\) is a solution to the equation \(d y / d x=g(x) .\) If the statement is true, explain how you know. If the
View solution Problem 42
Solve the differential equations in Problems \(34-43 .\) Assume \(a, b,\) and \(k\) are nonzero constants. $$\frac{d R}{d x}=a\left(R^{2}+1\right)$$
View solution