Inverse Trigonometric Functions

We are given the function: and asked about its sign-changing behavior in the interval .

Let us analyze the behavior of as approaches 0 from the left:

• As , we have .
• So, .

Now consider the composition . Since becomes large in magnitude and negative near , the input to the sine function becomes a large negative number.

We know that the sine function, , is periodic and oscillates between and . It crosses zero at multiples of :

when for any integer .

So, as , passes through infinitely many multiples of in the negative direction. That means:

• will be for infinitely many values of .
• Between any two zeros of the sine function, it takes on both positive and negative values.

Therefore, changes sign infinitely many times as .

Since never equals (we're only looking in the interval ), this oscillation continues endlessly as we approach from the left.

Final conclusion: The function changes sign infinitely many times in the interval .

Correct option: (D) changes sign infinitely many times