Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
g(x) = x sin⁻¹ x on [-1, 1]

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First, understand that critical points occur where the derivative of the function is zero or undefined. We need to find the derivative of g(x) = x sin⁻¹ x.

To find the derivative, apply the product rule: if u(x) = x and v(x) = sin⁻¹ x, then g'(x) = u'(x)v(x) + u(x)v'(x).

Calculate u'(x) = 1 and v'(x) = 1/√(1-x²) using the derivative of sin⁻¹ x. Substitute these into the product rule formula to get g'(x) = sin⁻¹ x + x/√(1-x²).

Set g'(x) = 0 to find critical points: sin⁻¹ x + x/√(1-x²) = 0. Solve this equation for x within the interval [-1, 1].

Evaluate g(x) at the critical points and endpoints of the interval [-1, 1] to determine the absolute maximum and minimum values. Compare these values to identify the absolute extrema.

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Critical Points

Critical points of a function occur where its derivative is either zero or undefined. These points are essential for identifying local maxima and minima, as they represent potential locations where the function's behavior changes. To find critical points, one must first compute the derivative of the function and solve for the values of x that satisfy these conditions.

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The inverse sine function, denoted as sin⁻¹(x) or arcsin(x), is the function that returns the angle whose sine is x. It is defined for x in the range [-1, 1] and outputs angles in the range [-π/2, π/2]. Understanding the properties of the inverse sine function is crucial when analyzing the function g(x) = x sin⁻¹(x), as it affects the behavior and differentiability of g(x) within the specified interval.

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