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The function f(x) = | x - 2| + | 2.5 - x| + |3.6 - x|, where x is a real number, attains a minimum at
- a)x=2.3
- b)x=2.5
- c)x=2.4
- d)None of the above
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The function f(x) = | x - 2| + | 2.5 - x| + |3.6 - x|, where x is a re...
The best way of doing this question is to substitute the value of x in the option to the given expression in f(x).
We get the minimum value of 1.6 for f(x) when x=2.5.
Now, there is a question whether the minimum value of the expression can be lesser than this.
In order to resolve the ambiguity,plot f(x) vs x in the graph for x=1,2,3,4 and so on.
We find that the minimum value is acheived at x=2.5.
Hence option(2) is the answer.
Most Upvoted Answer
The function f(x) = | x - 2| + | 2.5 - x| + |3.6 - x|, where x is a re...
Solution:
To find the minimum value of the given function, we need to find the critical points of the function.
Critical points are the points where the derivative of the function is zero or the derivative does not exist.
Let's find the derivative of the given function.
f(x) = |x - 2| |2.5 - x| |3.6 - x|
f'(x) = [(x - 2) / |x - 2|] |2.5 - x| |3.6 - x| + |x - 2| [(2.5 - x) / |2.5 - x|] |3.6 - x| + |x - 2| |2.5 - x| [(x - 3.6) / |3.6 - x|]
Now, to find the critical points, we need to set f'(x) = 0 and solve for x.
[(x - 2) / |x - 2|] |2.5 - x| |3.6 - x| + |x - 2| [(2.5 - x) / |2.5 - x|] |3.6 - x| + |x - 2| |2.5 - x| [(x - 3.6) / |3.6 - x|] = 0
We can simplify this equation by dividing both sides by the common factor |x - 2| |2.5 - x| |3.6 - x|.
[(x - 2) / |x - 2|] + [(2.5 - x) / |2.5 - x|] + [(x - 3.6) / |3.6 - x|] = 0
Now, we can solve this equation by considering the three cases:
Case 1: x < />
In this case, we have:
[(x - 2) / (2 - x)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (x - 3.6)] = 0
Simplifying this equation, we get:
x = 2.4
Case 2: 2.5 < x="" />< />
In this case, we have:
[(x - 2) / (x - 2)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (x - 3.6)] = 0
This equation is not defined at x = 2.5 and x = 3.6. However, we can see that the function is continuous in this interval, and it attains its minimum value at x = 2.5. Therefore, x = 2.5 is a critical point.
Case 3: x > 3.6
In this case, we have:
[(x - 2) / (x - 2)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (3.6 - x)] = 0
Simplifying this equation, we get:
x = 3
Therefore, the critical points of the function are x
To find the minimum value of the given function, we need to find the critical points of the function.
Critical points are the points where the derivative of the function is zero or the derivative does not exist.
Let's find the derivative of the given function.
f(x) = |x - 2| |2.5 - x| |3.6 - x|
f'(x) = [(x - 2) / |x - 2|] |2.5 - x| |3.6 - x| + |x - 2| [(2.5 - x) / |2.5 - x|] |3.6 - x| + |x - 2| |2.5 - x| [(x - 3.6) / |3.6 - x|]
Now, to find the critical points, we need to set f'(x) = 0 and solve for x.
[(x - 2) / |x - 2|] |2.5 - x| |3.6 - x| + |x - 2| [(2.5 - x) / |2.5 - x|] |3.6 - x| + |x - 2| |2.5 - x| [(x - 3.6) / |3.6 - x|] = 0
We can simplify this equation by dividing both sides by the common factor |x - 2| |2.5 - x| |3.6 - x|.
[(x - 2) / |x - 2|] + [(2.5 - x) / |2.5 - x|] + [(x - 3.6) / |3.6 - x|] = 0
Now, we can solve this equation by considering the three cases:
Case 1: x < />
In this case, we have:
[(x - 2) / (2 - x)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (x - 3.6)] = 0
Simplifying this equation, we get:
x = 2.4
Case 2: 2.5 < x="" />< />
In this case, we have:
[(x - 2) / (x - 2)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (x - 3.6)] = 0
This equation is not defined at x = 2.5 and x = 3.6. However, we can see that the function is continuous in this interval, and it attains its minimum value at x = 2.5. Therefore, x = 2.5 is a critical point.
Case 3: x > 3.6
In this case, we have:
[(x - 2) / (x - 2)] + [(2.5 - x) / (2.5 - x)] + [(x - 3.6) / (3.6 - x)] = 0
Simplifying this equation, we get:
x = 3
Therefore, the critical points of the function are x
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