Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > In the interval [0,π] the equation x = c...
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In the interval [0,π] the equation x = cos x has
- a)No solution
- b)Exactly one solution
- c)Exactly two solutions
- d)An infinite number of solutions
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In the interval [0,π] the equation x = cos x hasa)No solutionb)Ex...
f you consider x=0 then cosx=1
now if x= PI/4 = 0.785 then cosx=0.7071
for some x value x=cosx
after this x is increasing and cosx is decreasing. so we can say exactly 1 solution.
now if x= PI/4 = 0.785 then cosx=0.7071
for some x value x=cosx
after this x is increasing and cosx is decreasing. so we can say exactly 1 solution.
EDIT-
It is very easy to show that the equation x = cos x has a unique solution. For example take f(x) = c - cos x and notice that
(equality holding in isolated points) so f(x) is strictly increasing and hence the equation can have at most one solution.
It is very easy to show that the equation x = cos x has a unique solution. For example take f(x) = c - cos x and notice that
(equality holding in isolated points) so f(x) is strictly increasing and hence the equation can have at most one solution.At
and function is continious (difference of two continuous functions is continuous). Therefore there is solution in ,
hence there is a solution in [0,π]
and function is continious (difference of two continuous functions is continuous). Therefore there is solution in , hence there is a solution in [0,π]Most Upvoted Answer
In the interval [0,π] the equation x = cos x hasa)No solutionb)Ex...
Introduction:
We are given the equation x = cos x and we need to determine the number of solutions in the interval [0,].
Explanation:
To solve this equation, we can graph the functions y = x and y = cos x and find the points of intersection.
Graphical Representation:
Let's plot the graphs of y = x and y = cos x in the same coordinate system.
- The graph of y = x is a straight line passing through the origin with a slope of 1.
- The graph of y = cos x is a periodic function with an amplitude of 1 and a period of 2. It oscillates between -1 and 1.
Observations:
By observing the graph, we can make the following observations:
1. Intersection points between y = x and y = cos x will correspond to the solutions of the equation x = cos x.
2. There is one obvious intersection point at x = 0.
3. As x increases, the value of cos x decreases.
4. At x = , the value of cos x is approximately -0.739.
5. After x = , the value of cos x becomes less than -1, which means there are no more intersection points in the interval [0,].
Conclusion:
From the graphical representation and observations, we can conclude that there is exactly one solution to the equation x = cos x in the interval [0,].
Therefore, the correct answer is option 'B' - Exactly one solution.
We are given the equation x = cos x and we need to determine the number of solutions in the interval [0,].
Explanation:
To solve this equation, we can graph the functions y = x and y = cos x and find the points of intersection.
Graphical Representation:
Let's plot the graphs of y = x and y = cos x in the same coordinate system.
- The graph of y = x is a straight line passing through the origin with a slope of 1.
- The graph of y = cos x is a periodic function with an amplitude of 1 and a period of 2. It oscillates between -1 and 1.
Observations:
By observing the graph, we can make the following observations:
1. Intersection points between y = x and y = cos x will correspond to the solutions of the equation x = cos x.
2. There is one obvious intersection point at x = 0.
3. As x increases, the value of cos x decreases.
4. At x = , the value of cos x is approximately -0.739.
5. After x = , the value of cos x becomes less than -1, which means there are no more intersection points in the interval [0,].
Conclusion:
From the graphical representation and observations, we can conclude that there is exactly one solution to the equation x = cos x in the interval [0,].
Therefore, the correct answer is option 'B' - Exactly one solution.
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In the interval [0,π] the equation x = cos x hasa)No solutionb)Exactly one solutionc)Exactly two solutionsd)An infinite number of solutionsCorrect answer is option 'B'. Can you explain this answer?
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