In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.

g(x) = { x²/³, x ≥ 0
x¹/³, x < 0

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First, understand that a function is differentiable at a point if it is continuous at that point and if the derivative exists at that point.

Check the continuity of the function at x = 0. Evaluate the left-hand limit and the right-hand limit of g(x) as x approaches 0. For x ≥ 0, g(x) = x^(2/3), and for x < 0, g(x) = x^(1/3).

Calculate the left-hand limit: lim (x -> 0-) g(x) = lim (x -> 0-) x^(1/3).

Calculate the right-hand limit: lim (x -> 0+) g(x) = lim (x -> 0+) x^(2/3).

Determine if the derivative exists at x = 0 by finding the left-hand derivative and the right-hand derivative at x = 0. For x ≥ 0, find the derivative of x^(2/3), and for x < 0, find the derivative of x^(1/3). Compare these derivatives at x = 0.

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주요 개념

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Differentiability

Differentiability at a point means that the function has a defined derivative at that point. For a function to be differentiable at x = 0, it must be continuous at x = 0, and the left-hand and right-hand derivatives at x = 0 must exist and be equal. If these conditions are not met, the function is not differentiable at that point.

Piecewise-Defined Functions

A piecewise-defined function is a function composed of multiple sub-functions, each applying to a certain interval of the domain. To analyze differentiability at a point where the definition changes, like x = 0 in this case, we must consider the behavior of each sub-function at that point and ensure they connect smoothly.

Continuity and Limits

Continuity at a point requires that the limit of the function as it approaches the point from both sides equals the function's value at that point. For the given piecewise function, we must check if the limits of x²/³ as x approaches 0 from the right and x¹/³ as x approaches 0 from the left both equal g(0). This is a prerequisite for differentiability.

In Exercises 41–44, determine whether the piecewise-defined function is differentiable at x = 0.g(x) = { x²/³, x ≥ 0 x¹/³, x < 0