Let be a continuous function on such that its derivative is p'(x)= x^ 2 -4 (x^ 2 -1 )^ 2/3

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

Problem

Let

p

be a continuous function on

\mathbb R

such that its derivative is

p'(x)=\frac{x^{2}-4}{\bigl(x^{2}-1\bigr)^{2/3}}, \qquad x\neq\pm1.

Assume no further restrictions on

p

. (a) Identify all real numbers where

p'

is not defined. For each such number

c

, determine whether

p'(x)

diverges to

\infty

, diverges to

-\infty

, or approaches finite (possibly unequal) one-sided limits as

x\to c

. (b) Using limits of

p'(x)

, decide at which of the points found in part (a) the function

p

is not differentiable. Support your answer with appropriate limit statements. (c) For every

x

-value where

p

is not differentiable, classify the specific feature present on the graph of

p

(corner, cusp, vertical tangent, or a point where

p

itself is discontinuous). Provide detailed justification referencing the precise limiting behavior of

p'

. (d) Suppose additionally that

p(0)=3

and

p'(x)>0

for

x>2

. Describe the qualitative shape of

p

for

x>2

and explain how the non-differentiable behavior discovered in part (c) influences the overall graph.

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