A function g is defined by g(x)=secx on the interval (−2π,23π).
Answer the following questions, supporting all claims with appropriate calculus justification.
(a) Find g′′(x). Show algebraic steps leading to your result.
(b) Determine all intervals on which the graph of g is concave upward and all intervals on which it is concave downward within the stated domain.
(c) Identify every point of inflection of g in (−2π,23π) and verify that the concavity actually changes at each point you list.
(d) Describe how the concavity information from part (b) could be used to improve a second–degree Taylor polynomial approximation to secx centered at x=0. Give a qualitative, sentence–level explanation rather than a full derivation.