Chapter 4: Problem 13
In Exercises \(13-20\) , find all points of inflection of the function. $$y=x e^{x}$$
Short Answer
Expert verified
The points of inflection of the function \(y=x e^{x}\) are (-2,-0.368) and (0,0).
Step by step solution
01
Differentiate
Find the first derivative of the function \(y = xe^{x}\). Utilize the product rule, which states that the derivative of the product of two functions is the first function times the derivative of the second, plus the second function times the derivative of the first. Thus, \(y' = e^{x} + xe^{x}\).
02
Differentiate again
Determine the second derivative by differentiating the derivative result using the product rule again. This yields: \(y'' = e^{x} + e^{x} + xe^{x} = 2e^{x} + xe^{x}\).
03
Find critical points
To find the inflection points, set the second derivative equal to zero, and solve for x: \(0 = 2e^{x} + xe^{x}.\) This simplifies to: \(0 = e^{x}(2+x)\). Thus, x = -2 and x = 0 are the solutions.
04
Test for inflection
An inflection point makes a sign change in the second derivative on either side. Calculate \(y''\) for values to either side of x=0 and x=-2 to confirm they are indeed points of inflection. We find \(y''(-3)<0\), \(y''(-1)>0\), \(y''(-0.5)<0\), and \(y''(0.5)>0\), confirming x = -2 and x = 0 as points of inflection.
05
Find the y-coordinate
Finally, find the y-coordinates of the points of inflection by substituting x = -2 and x = 0 into the original equation \(y = xe^{x}\). So the points of inflection are (-2,-0.368) and (0,0).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Product Rule
Understanding the product rule is essential when differentiating functions that are products of two or more functions. Like in our exercise, where we have to differentiate the function y = xe^{x}, the product rule comes into play. According to the product rule, if you have two functions u(x) and v(x), their derivative u'(x)v(x) + u(x)v'(x) is a combination of each function multiplied by the derivative of the other.
Applying this to the function xe^{x}, we see it's a product of x (which is u(x) in this context) and e^{x} (which is v(x)). Differentiating x gives us 1, and the derivative of e^{x} is e^{x}; when we apply the product rule, we get the first derivative y' = e^{x} + xe^{x}. It's important to apply this rule systematically to ensure accuracy and establish a foundation for further differentiation, as we'll do in the following steps.
Applying this to the function xe^{x}, we see it's a product of x (which is u(x) in this context) and e^{x} (which is v(x)). Differentiating x gives us 1, and the derivative of e^{x} is e^{x}; when we apply the product rule, we get the first derivative y' = e^{x} + xe^{x}. It's important to apply this rule systematically to ensure accuracy and establish a foundation for further differentiation, as we'll do in the following steps.
Second Derivative Test
The second derivative test is a handy tool for determining if a point on a graph is a local maximum, minimum, or neither. The test involves taking the second derivative of the function, as we've done with y'' = 2e^{x} + xe^{x}.
If the second derivative at a certain point is positive, the function is concave up at that point, suggesting a local minimum. If the second derivative is negative, the function is concave down, which usually indicates a local maximum. However, when the second derivative equals zero, it can be a candidate for an inflection point; this is where the behavior of the curve changes from concave up to concave down or vice versa. In our exercise, we set the second derivative equal to zero to find potential inflection points and test the intervals around them to confirm the change in concavity, ensuring we've located true points of inflection.
If the second derivative at a certain point is positive, the function is concave up at that point, suggesting a local minimum. If the second derivative is negative, the function is concave down, which usually indicates a local maximum. However, when the second derivative equals zero, it can be a candidate for an inflection point; this is where the behavior of the curve changes from concave up to concave down or vice versa. In our exercise, we set the second derivative equal to zero to find potential inflection points and test the intervals around them to confirm the change in concavity, ensuring we've located true points of inflection.
Inflection Point Calculus
Inflection point calculus revolves around finding where the concavity of a function changes. An inflection point is where a curve switches from being concave upwards (cup-shaped) to concave downwards (cap-shaped), or vice versa. To find a function's points of inflection, we first find its second derivative and then determine where this derivative is zero or undefined, which are our potential inflection points.
In our exercise, after finding the second derivative y'' = 2e^{x} + xe^{x}, we solve 0 = 2e^{x} + xe^{x} to find where the second derivative may change sign. It's not enough for the second derivative to be zero; the sign must change on either side of the point. By substituting values into the second derivative just smaller and larger than our candidates, we confirmed that the sign does indeed change around x = -2 and x = 0, thus verifying these as inflection points.
In our exercise, after finding the second derivative y'' = 2e^{x} + xe^{x}, we solve 0 = 2e^{x} + xe^{x} to find where the second derivative may change sign. It's not enough for the second derivative to be zero; the sign must change on either side of the point. By substituting values into the second derivative just smaller and larger than our candidates, we confirmed that the sign does indeed change around x = -2 and x = 0, thus verifying these as inflection points.
Differentiation
Differentiation is the process by which we find the derivative, or rate of change, of a function. It is a fundamental tool in calculus that allows us to understand how a function varies at any point along its curve. The derivative tells us the slope of the tangent line to the curve at a given point, which corresponds to the instantaneous rate of change.
In the exercise involving the function y = xe^{x}, differentiation is used twice. The first derivative, found using the product rule, shows us how the function changes with respect to x. Taking the derivative of the derivative—called the second derivative—allows us to determine the function's concavity and find its inflection points. Differentiation can be applied in various ways, depending on the function and what we seek to understand about its behavior.
In the exercise involving the function y = xe^{x}, differentiation is used twice. The first derivative, found using the product rule, shows us how the function changes with respect to x. Taking the derivative of the derivative—called the second derivative—allows us to determine the function's concavity and find its inflection points. Differentiation can be applied in various ways, depending on the function and what we seek to understand about its behavior.
