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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?
Most Upvoted Answer
The function f(x) = cot1 x + x increases in the intervala)(1, ∞)...
The function f(x) = cot1 x increases in the interval (1, pi/2).
To see this, let's first recall the definition of the cotangent function. cot(x) = 1/tan(x) = cos(x)/sin(x).
Now, let's consider the interval (0, pi/2). In this interval, both the cosine and sine functions are positive, so cot(x) = cos(x)/sin(x) is also positive.
Next, let's take the derivative of cot(x). Using the quotient rule, we get:
f'(x) = -csc^2(x)
Since the derivative is negative for all x in the interval (0, pi/2), this means that the function is decreasing in this interval.
However, if we look at the interval (1, pi/2), we can see that the function is still positive, but now the derivative is positive as well. This means that the function is increasing in this interval. Therefore, we can conclude that the function f(x) = cot1 x increases in the interval (1, pi/2).
To see this, let's first recall the definition of the cotangent function. cot(x) = 1/tan(x) = cos(x)/sin(x).
Now, let's consider the interval (0, pi/2). In this interval, both the cosine and sine functions are positive, so cot(x) = cos(x)/sin(x) is also positive.
Next, let's take the derivative of cot(x). Using the quotient rule, we get:
f'(x) = -csc^2(x)
Since the derivative is negative for all x in the interval (0, pi/2), this means that the function is decreasing in this interval.
However, if we look at the interval (1, pi/2), we can see that the function is still positive, but now the derivative is positive as well. This means that the function is increasing in this interval. Therefore, we can conclude that the function f(x) = cot1 x increases in the interval (1, pi/2).
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The function f(x) = cot1 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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