Let h(x)= cases x^ 23 , & x<0

MATHEMATICS · Multi-part · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Let h(x)={x23,x<0,kx+1,0≤x<2,4−x,x≥2,h(x)=\begin{cases} x^{\tfrac23}, & x<0,\\ kx+1, & 0\le x<2,\\ \sqrt{4-x}, & x\ge2, \end{cases} and define g(x)=x2 h(x)g(x)=x^2\,h(x). Answer the following questions. (1) Part 1: (a) Determine the value of the constant kk that makes hh continuous on (−∞,∞)(-\infty,\infty). Show algebraic work. (2) Part 2: (b) Identify all xx–values where hh is not differentiable. For each such xx–value, classify the nondifferentiability as a corner, cusp, or vertical tangent, and support each classification by evaluating the appropriate one-sided derivatives. (3) Part 3: (c) Using your conclusions from part (b), determine all xx–values where gg is not differentiable. Explain how multiplying by x2x^2 affects (or does not affect) the locations and classifications of the points of nondifferentiability.

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