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Let f: R→R be a differentiable function with f(0) = 0. If for all x ∈ R, 1 < f'(x) < 2, then which one of the following statements is true on (0, ∝)?
- a)f is unbounded
- b)f is increasing and bounded
- c)f has at least one zero
- d)f is periodic
Correct answer is option 'A'. Can you explain this answer?
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Let f: R→R be a differentiable function with f(0) = 0. If for al...
Answer:
To determine the correct statement about the function f(x), we need to analyze the given conditions and properties of the function.
Condition 1: f(0) = 0
This condition tells us that the function passes through the point (0,0).
Condition 2: 1 f(x) 2 for all x in R
This condition tells us that the range of the function lies between 1 and 2 for all values of x in the domain.
Statement a: f is unbounded
If f is unbounded, it means that the function has no upper or lower limit as x approaches infinity. In other words, the function grows without bound or decreases without bound.
To determine if this statement is true, we need to examine the behavior of the function as x approaches infinity.
Since the range of f(x) is bounded between 1 and 2 for all x, the function cannot be unbounded. Thus, statement a is false.
Statement b: f is increasing and bounded
If f is increasing, it means that the function values increase as x increases. If f is bounded, it means that the function has an upper and lower limit.
Since the range of f(x) is bounded between 1 and 2 for all x, the function is bounded. However, we cannot determine if the function is increasing or decreasing based on the given information. Thus, statement b is false.
Statement c: f has at least one zero
Since f(0) = 0, we know that the function has at least one zero. Thus, statement c is true.
Statement d: f is periodic
A periodic function repeats its values in a regular pattern. Since the range of f(x) is bounded between 1 and 2 for all x, the function cannot repeat its values in a regular pattern. Thus, statement d is false.
Therefore, the correct statement is option 'A': f is unbounded.
To determine the correct statement about the function f(x), we need to analyze the given conditions and properties of the function.
Condition 1: f(0) = 0
This condition tells us that the function passes through the point (0,0).
Condition 2: 1 f(x) 2 for all x in R
This condition tells us that the range of the function lies between 1 and 2 for all values of x in the domain.
Statement a: f is unbounded
If f is unbounded, it means that the function has no upper or lower limit as x approaches infinity. In other words, the function grows without bound or decreases without bound.
To determine if this statement is true, we need to examine the behavior of the function as x approaches infinity.
Since the range of f(x) is bounded between 1 and 2 for all x, the function cannot be unbounded. Thus, statement a is false.
Statement b: f is increasing and bounded
If f is increasing, it means that the function values increase as x increases. If f is bounded, it means that the function has an upper and lower limit.
Since the range of f(x) is bounded between 1 and 2 for all x, the function is bounded. However, we cannot determine if the function is increasing or decreasing based on the given information. Thus, statement b is false.
Statement c: f has at least one zero
Since f(0) = 0, we know that the function has at least one zero. Thus, statement c is true.
Statement d: f is periodic
A periodic function repeats its values in a regular pattern. Since the range of f(x) is bounded between 1 and 2 for all x, the function cannot repeat its values in a regular pattern. Thus, statement d is false.
Therefore, the correct statement is option 'A': f is unbounded.
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Let f: R→R be a differentiable function with f(0) = 0. If for all x∈ R, 1 < f'(x) < 2, then which one of the following statements is true on (0, ∝)?a)f is unboundedb)f is increasing and boundedc)f has at least one zerod)f is periodicCorrect answer is option 'A'. Can you explain this answer?
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