To determine the number of values of \( x \) in the interval \([-1, 1]\) where the function \( f(x) = \sin^{-1}(\min(|x|, |y|)) \) is not differentiable, we first need to analyze the components of the function and the curve defined by \( y = \sqrt{1 - x^2} \).
Understanding the Function and the Curve
The function \( y = \sqrt{1 - x^2} \) represents the upper half of a circle with a radius of 1, centered at the origin. This means that for any \( x \) in the interval \([-1, 1]\), \( y \) will take values from 0 to 1. The absolute values \( |x| \) and \( |y| \) will also be in the range \([0, 1]\).
Identifying the Points of Interest
The function \( f(x) \) is defined as the inverse sine of the minimum of \( |x| \) and \( |y| \). Therefore, we need to find where the minimum function changes, which occurs at the points where \( |x| = |y| \).
- Setting \( |x| = |y| \) gives us \( |x| = \sqrt{1 - x^2} \).
- Squaring both sides leads to \( x^2 = 1 - x^2 \), simplifying to \( 2x^2 = 1 \), or \( x^2 = \frac{1}{2} \).
- This results in \( x = \pm \frac{1}{\sqrt{2}} \) or approximately \( \pm 0.707 \).
Analyzing Differentiability
To determine where \( f(x) \) is not differentiable, we need to consider the points where the minimum function changes from \( |x| \) to \( |y| \). This occurs at the points we calculated above:
- At \( x = \frac{1}{\sqrt{2}} \), \( f(x) \) transitions from being dependent on \( |y| \) to \( |x| \).
- At \( x = -\frac{1}{\sqrt{2}} \), a similar transition occurs.
At these points, the derivative of \( f(x) \) may not exist because the function switches from one expression to another, leading to a potential corner or cusp in the graph.
Conclusion on Non-Differentiability
Thus, the function \( f(x) \) is not differentiable at two specific points within the interval \([-1, 1]\): \( x = \frac{1}{\sqrt{2}} \) and \( x = -\frac{1}{\sqrt{2}} \). Therefore, the total number of values of \( x \) in the specified interval where \( f(x) \) is not differentiable is:
2
Visual Representation
While I can't provide a detailed graph directly here, you can visualize the scenario as follows:
- Sketch the circle defined by \( y = \sqrt{1 - x^2} \).
- Mark the points \( x = \frac{1}{\sqrt{2}} \) and \( x = -\frac{1}{\sqrt{2}} \) on the x-axis.
- Observe how the function \( f(x) \) behaves around these points, noting the changes in the minimum function.
By plotting these elements, you will see the transitions that indicate non-differentiability at the specified points. This visual aid can greatly enhance your understanding of the function's behavior in relation to the curve.
