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Can you explain the answer of this question below:
The function f(x) = cot⁻1x + x increases in the interval
- A:(1, ∞)
- B:(-1, ∞)
- C:(-∞, ∞)
- D:(0, ∞)
- E:undefined
The answer is c.
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Can you explain the answer of this question below:The function f(x) = ...
Explanation:
Interval Analysis:
- The function f(x) = cot(1x) + x is increasing if its derivative is positive in the given interval.
Derivative of f(x):
- To find the derivative of f(x), we need to differentiate cot(1x) and x separately.
- Derivative of cot(1x) is -csc^2(1x) * 1 = -csc^2(x).
- Derivative of x is 1.
Derivative of f(x):
- Therefore, the derivative of f(x) = cot(1x) + x is -csc^2(x) + 1.
Analysis of Derivative:
- The derivative -csc^2(x) + 1 is positive for all real numbers x.
- The function f(x) = cot(1x) + x is increasing for all real numbers x.
- Hence, the function increases in the interval (-∞, ∞).
Therefore, the correct answer is option C (-∞, ∞).
Interval Analysis:
- The function f(x) = cot(1x) + x is increasing if its derivative is positive in the given interval.
Derivative of f(x):
- To find the derivative of f(x), we need to differentiate cot(1x) and x separately.
- Derivative of cot(1x) is -csc^2(1x) * 1 = -csc^2(x).
- Derivative of x is 1.
Derivative of f(x):
- Therefore, the derivative of f(x) = cot(1x) + x is -csc^2(x) + 1.
Analysis of Derivative:
- The derivative -csc^2(x) + 1 is positive for all real numbers x.
- The function f(x) = cot(1x) + x is increasing for all real numbers x.
- Hence, the function increases in the interval (-∞, ∞).
Therefore, the correct answer is option C (-∞, ∞).
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Can you explain the answer of this question below:The function f(x) = cot⁻1x + x increases in the intervalA:(1, ∞)B:(-1, ∞)C:(-∞, ∞)D:(0, ∞)E:undefinedThe answer is c.
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