For a real parameter

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

For a real parameter aa, consider the piecewise function fa(x)={ax+x2,x≤0,x,x>0.\boxed{f_a(x)=\begin{cases} ax+x^{2}, & x\le 0,\\ \sqrt{x}, & x>0. \end{cases}} The value of fa(0)f_a(0) is determined by the first branch. (1) Part 1: Determine all real values of aa for which faf_a is continuous at x=0x=0. Show all work. (2) Part 2: For each value of aa found in part (a), evaluate the left–hand and right–hand derivatives of faf_a at x=0x=0. Decide whether faf_a is differentiable at x=0x=0 and support your reasoning with appropriate limits. (3) Part 3: Classify the type of non-differentiability you find in part (b) (corner, cusp, or vertical tangent) and explain your classification using the behavior of the derivative limits.

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