Let f(x)=(x^ 2 -9)^ 1/3 be defined for all real .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Definition and Fundamental Properties · Differentiation: Definition and Fundamental Properties

Problem

Let f(x)=(x2−9)1/3f(x)=(x^{2}-9)^{1/3} be defined for all real xx. (a) For x≠±3x\neq\pm 3, compute f′(x)f'(x) by differentiating an appropriate algebraic form of ff. (b) Evaluate lim⁡x→3−f′(x)\displaystyle\lim_{x\to 3^-}f'(x) and lim⁡x→3+f′(x)\displaystyle\lim_{x\to 3^+}f'(x), showing all work. (c) Determine whether ff is differentiable at x=3x=3. Justify fully, commenting on both continuity and the behavior of the derivative. (d) Describe the appearance of the graph of y=f(x)y=f(x) near x=3x=3, including the slope of any tangent line that may exist.

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