Consider the function defined on the interval .

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

Problem

Consider the function

f(x)=\tan x- x

defined on the interval

\left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)

. (a) Compute

f'(x)

and identify every value of

x

in the given interval where

f'(x)

is zero or undefined. (b) Use your work from part (a) to determine all open intervals on which

f

is increasing and all open intervals on which

f

is decreasing. Provide clear reasoning tied to the sign of

f'(x)

. (c) Evaluate the second derivative

f''(x)

and identify all values of

x

in the given interval where

f''(x)

is zero or undefined. (d) Using the information from parts (a)–(c), describe all intervals of concavity for

f

and locate every point of inflection within the interval. Justify each claim with appropriate reasoning.