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f (x) is a continuous function and takes only rational values. If f (0) = 3, then f (2) equals
- a)5
- b)0
- c)1
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
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f (x) is a continuous function and takes only rational values. If f (0...
Solution:
Given, f(x) is a continuous function and takes only rational values. Also, f(0) = 3.
To find: f(2)
Assume that f(x) is a polynomial function of degree n.
Then, f(x) = anxn + an-1xn-1 + … + a1x + a0, where a0, a1, …, an are rational constants.
Since f(x) takes only rational values, all coefficients are rational.
Also, f(0) = a0 = 3.
Now, consider the interval [0,2].
By the intermediate value theorem, f(x) takes all values between f(0) = 3 and f(2) in this interval.
Since f(x) takes only rational values, f(2) must also be rational.
However, there is no polynomial function with rational coefficients that passes through the points (0,3) and (2, irrational).
Therefore, such a function f(x) cannot exist.
Hence, the correct answer is option D, None of these.
Given, f(x) is a continuous function and takes only rational values. Also, f(0) = 3.
To find: f(2)
Assume that f(x) is a polynomial function of degree n.
Then, f(x) = anxn + an-1xn-1 + … + a1x + a0, where a0, a1, …, an are rational constants.
Since f(x) takes only rational values, all coefficients are rational.
Also, f(0) = a0 = 3.
Now, consider the interval [0,2].
By the intermediate value theorem, f(x) takes all values between f(0) = 3 and f(2) in this interval.
Since f(x) takes only rational values, f(2) must also be rational.
However, there is no polynomial function with rational coefficients that passes through the points (0,3) and (2, irrational).
Therefore, such a function f(x) cannot exist.
Hence, the correct answer is option D, None of these.
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f (x) is a continuous function and takes only rational values. If f (0...
If f(x) is a continuous function and takes only rational values then f(x) is a constant function. f(0) = 3 then f(2) = 3 . so answer is none of these
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f (x) is a continuous function and takes only rational values. If f (0) = 3, then f (2) equalsa)5b)0c)1d)None of theseCorrect answer is option 'D'. Can you explain this answer?
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