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The equation x log x = 3 - x has, in the interval (1, 3),
- a)Exactly one root
- b)Atmost one root
- c)Atleast one root
- d)No root
Correct answer is option 'C'. Can you explain this answer?
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The equation x log x = 3 - x has, in the interval (1, 3),a)Exactly one...
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The equation x log x = 3 - x has, in the interval (1, 3),a)Exactly one...
Given Equation:
The given equation is x log x = 3 - x.
Simplifying the Equation:
To solve the equation, we can start by simplifying it.
Using logarithmic rules, we can rewrite the equation as:
log x^x = 3 - x
Graphical Representation:
To understand the behavior of the equation in the interval (1, 3), we can plot the graph of the equation.
Drawing the Graph:
- Plot the graph of f(x) = x log x.
- Plot the graph of g(x) = 3 - x.
- Observe the intersection points of the two graphs.
Interpreting the Graph:
From the graph, we can observe the following:
- The graph of f(x) = x log x is continuous in the interval (1, 3).
- The graph of g(x) = 3 - x is also continuous in the interval (1, 3).
- The two graphs intersect at least once in the interval (1, 3).
Conclusion:
Based on the graphical representation and analysis, we can conclude that the equation x log x = 3 - x has at least one root in the interval (1, 3). Therefore, the correct answer is option 'c' - At least one root.
The given equation is x log x = 3 - x.
Simplifying the Equation:
To solve the equation, we can start by simplifying it.
Using logarithmic rules, we can rewrite the equation as:
log x^x = 3 - x
Graphical Representation:
To understand the behavior of the equation in the interval (1, 3), we can plot the graph of the equation.
Drawing the Graph:
- Plot the graph of f(x) = x log x.
- Plot the graph of g(x) = 3 - x.
- Observe the intersection points of the two graphs.
Interpreting the Graph:
From the graph, we can observe the following:
- The graph of f(x) = x log x is continuous in the interval (1, 3).
- The graph of g(x) = 3 - x is also continuous in the interval (1, 3).
- The two graphs intersect at least once in the interval (1, 3).
Conclusion:
Based on the graphical representation and analysis, we can conclude that the equation x log x = 3 - x has at least one root in the interval (1, 3). Therefore, the correct answer is option 'c' - At least one root.
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The equation x log x = 3 - x has, in the interval (1, 3),a)Exactly one rootb)Atmost one rootc)Atleast one rootd)No rootCorrect answer is option 'C'. Can you explain this answer?
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