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The function f (x) = cot-1 x+ x increases in the interval
- a)(1, ∞)
- b)(-1,∞)
- c)(-∞, ∞)
- d)(0, ∞)
Correct answer is option 'C'. Can you explain this answer?
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The function f (x) = cot-1 x+ x increases in the intervala)(1, ∞...
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The function f (x) = cot-1 x+ x increases in the intervala)(1, ∞...
Answer:
The function f(x) = cot^(-1)(x) is the inverse cotangent function. To determine the interval in which the function increases, we need to analyze the behavior of the inverse cotangent function.
Properties of the inverse cotangent function:
1. The domain of the inverse cotangent function is (-∞, ∞).
2. The range of the inverse cotangent function is (0, π) U (π, 2π).
3. The inverse cotangent function is increasing in each of its branches, which means that as x increases, cot^(-1)(x) also increases.
Analysis:
1. The function f(x) = cot^(-1)(x) is defined for all real numbers except x = 0, where cot^(-1)(x) is undefined.
2. As x approaches positive infinity, cot^(-1)(x) approaches 0. This is because the cotangent function approaches 0 as the angle approaches π/2.
3. As x approaches negative infinity, cot^(-1)(x) approaches π. This is because the cotangent function approaches 0 as the angle approaches -π/2, and adding π to the angle gives the cotangent value in the second quadrant, which is π.
4. Therefore, the range of the inverse cotangent function is (0, π), and it is increasing in this interval.
Conclusion:
The function f(x) = cot^(-1)(x) increases in the interval (0, π). Therefore, the correct answer is option 'C'.
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The function f (x) = cot-1 x+ x increases in the intervala)(1, ∞...
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The function f (x) = cot-1 x+ x increases in the intervala)(1, ∞)b)(-1,∞)c)(-∞, ∞)d)(0, ∞)Correct answer is option 'C'. Can you explain this answer?
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