Problem 138
Question
The equation \(x \log x=3-x\) has, in the interval \((1,3)\) (A) exactly one root (B) at least one root (C) at most one root (D) no root
Step-by-Step Solution
Verified Answer
(A) exactly one root
1Step 1: Understanding the Problem
We need to determine the number of roots of the equation \( x \log x = 3 - x \) within the interval \((1,3)\). This is a problem of finding where these two functions meet in the specified range.
2Step 2: Graphical Representation
Consider the functions \( f(x) = x \log x \) and \( g(x) = 3 - x \). We will analyze the behavior of these functions on the interval \((1,3)\). The function \( x \log x \) is increasing on this interval since the derivative \( \log x + 1 \) is positive for \( x > 1 \). The function \( g(x) = 3 - x \) is a linear function and decreases through \((1,3)\).
3Step 3: Finding the Intersection Points
To find out the points where these functions intersect, we set \( f(x) = g(x) \) or \( x \log x = 3 - x \). Examination of continuity and monotonic nature helps to infer potential intersections based on the behavior of the functions.
4Step 4: Analyzing Function Values at the Interval Ends
Calculate \( f(x) \) and \( g(x) \) at critical points within the interval. For instance, at \( x = 1 \), we have \( f(1) = 1 \log 1 = 0 \) and \( g(1) = 2 \). At \( x = 3 \), \( f(3) = 3 \log 3 \approx 3.295\) and \( g(3) = 0 \). \( g(x) \) starts above \( f(x) \) and ends below it.
5Step 5: Conclusion on Roots
Since \( f(x) = x \log x \) starts below \( g(x) = 3 - x \) at \( x = 1 \) and ends above \( g(x) \) at \( x = 3 \), and \( f(x) \) is increasing while \( g(x) \) is decreasing, these suggest there must be exactly one intersection in \( (1,3) \). Hence, there is exactly one root in this interval.
Key Concepts
Finding roots of an equationGraphical representation of functionsIntersection of functions
Finding roots of an equation
Finding the roots of an equation is the process of determining the values of the variable that satisfy the equation, making it equal to zero. In our context, we are interested in the points where two functions intersect, as this tells us where they are equal to each other.
In the given problem, the equation \( x \log x = 3 - x \) represents two functions: \( f(x) = x \log x \) and \( g(x) = 3 - x \). To find the roots, we must solve for \( x \) such that \( f(x) = g(x) \). This is equivalent to finding the "zero" of the difference between the functions, i.e., \( f(x) - g(x) = 0 \). Solving such equations often requires analytical methods or graphical visualization, especially when they involve transcendental functions like the logarithm.
In our specific example, since one function is strictly increasing and the other is strictly decreasing, there is a high possibility of having a single point of intersection, demonstrating a unique root where the values of both functions are equal.
In the given problem, the equation \( x \log x = 3 - x \) represents two functions: \( f(x) = x \log x \) and \( g(x) = 3 - x \). To find the roots, we must solve for \( x \) such that \( f(x) = g(x) \). This is equivalent to finding the "zero" of the difference between the functions, i.e., \( f(x) - g(x) = 0 \). Solving such equations often requires analytical methods or graphical visualization, especially when they involve transcendental functions like the logarithm.
In our specific example, since one function is strictly increasing and the other is strictly decreasing, there is a high possibility of having a single point of intersection, demonstrating a unique root where the values of both functions are equal.
Graphical representation of functions
Visualizing functions through graphs is an invaluable tool in understanding how functions behave over an interval.
Graphs can show you the overall trend of how a function increases or decreases and highlight possible intersection points with other functions. For the equation \( x \log x = 3 - x \), by plotting both \( x \log x \) and \( 3 - x \), you can observe their behavior over the interval \((1,3)\).
The function \( f(x) = x \log x \) is seen to increase as \( x \) increases in \((1,3)\). This is confirmed by checking its derivative, \( \log x + 1 \), which remains positive in this interval. On the other hand, \( g(x) = 3 - x \) is linear and decreases steadily across \((1,3)\), creating two opposite trends between these functions.
Such graphical representation quickly shows where one function begins below or above the other, helping determine the point(s) of intersection. In our exercise, the graphical approach aids in establishing that the functions start and finish in opposite order, indicating a crossing point, which signifies a root.
Graphs can show you the overall trend of how a function increases or decreases and highlight possible intersection points with other functions. For the equation \( x \log x = 3 - x \), by plotting both \( x \log x \) and \( 3 - x \), you can observe their behavior over the interval \((1,3)\).
The function \( f(x) = x \log x \) is seen to increase as \( x \) increases in \((1,3)\). This is confirmed by checking its derivative, \( \log x + 1 \), which remains positive in this interval. On the other hand, \( g(x) = 3 - x \) is linear and decreases steadily across \((1,3)\), creating two opposite trends between these functions.
Such graphical representation quickly shows where one function begins below or above the other, helping determine the point(s) of intersection. In our exercise, the graphical approach aids in establishing that the functions start and finish in opposite order, indicating a crossing point, which signifies a root.
Intersection of functions
When discussing the intersection of functions, we are examining the points at which two or more functions meet or have the same value. These intersection points are crucial because they often represent the solution or "roots" to equations formed by these functions.
In the provided task, finding intersections of \( f(x) = x \log x \) and \( g(x) = 3 - x \) is key to determining if there are any solutions within \( (1,3) \). The behavior of these functions, one being increasing and the other decreasing, means their graphs will cross exactly once in this interval.
At the point of intersection, both functions equal each other, and therefore \( x \log x = 3-x \) holds true. By examining these graphical trends and analytic observations, it's concluded that there is only a single point where this equality is satisfied within \( (1,3) \). Hence, exactly one intersection (and thus exactly one root) exists in the given range. By analyzing intersections, we comprehend not just where functions meet but also gain insights into the number of solutions present.
In the provided task, finding intersections of \( f(x) = x \log x \) and \( g(x) = 3 - x \) is key to determining if there are any solutions within \( (1,3) \). The behavior of these functions, one being increasing and the other decreasing, means their graphs will cross exactly once in this interval.
At the point of intersection, both functions equal each other, and therefore \( x \log x = 3-x \) holds true. By examining these graphical trends and analytic observations, it's concluded that there is only a single point where this equality is satisfied within \( (1,3) \). Hence, exactly one intersection (and thus exactly one root) exists in the given range. By analyzing intersections, we comprehend not just where functions meet but also gain insights into the number of solutions present.
Other exercises in this chapter
Problem 136
If \(a+b+c=0\), then the equation \(3 a x^{2}+2 b x+c=0\) has, in the interval \((0,1)\) (A) at least one root (B) at most one root (C) no root (D) None of thes
View solution Problem 137
If \(a+b+c=0\), then the equation \(3 a x^{2}+2 b x+c=0\) has, in the interval \((0,1)\) (A) at least one root (B) at most one root (C) no root (D) None of thes
View solution Problem 141
\(\frac{\sin \alpha-\sin \beta}{\cos \beta-\cos \alpha}=F(\alpha)\), where \(0
View solution Problem 143
$$ \begin{array}{ll} \text { Column-I } & \text { Column-II } \\ \hline \text { I. Let } f(x)=\left(1+b^{2}\right) x^{2}+2 b x+1 & \text { (A) }(0,1] \\ \text {
View solution