Let f be a differentiable function with derivative given by the expression Note that…

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let f be a differentiable function with derivative given by the expression

f'(x) = \frac{x^{2} - 4}{x - 1}, \qquad x \ne 1.

Note that

f'

is undefined at

x = 1

. No other discontinuities of

f'

occur, and

f

is continuous for all real numbers. Use this information to analyze the behavior of

f

. (a) Determine all intervals on which

f

is increasing and all intervals on which

f

is decreasing. (b) Identify every critical point of

f

and classify each as a relative maximum or a relative minimum. Provide a complete justification. (c) Describe the behavior of

f

near

x = 1

. Does the graph of

f

have a vertical tangent, a cusp, or neither at

x = 1

? Explain your reasoning. (d) Determine the intervals of concavity of

f

and locate each point of inflection. Justify your answers.