Mathematics

Problem

Consider the one-parameter family of functions defined on

(0,\pi)

F_k(x)=e^{x}+k\,\ln(\sin x),

where

k

is a real constant. a. Derive explicit formulas for

F_k'(x)

and

F_k''(x)

. Simplify your expressions as much as possible. b. Determine the value of

k

for which

F_k'\!\left(\dfrac{\pi}{4}\right)=0

. Provide exact numerical value(s). c. For the value of

k

obtained in part (b), decide whether

x=\dfrac{\pi}{4}

corresponds to a local maximum, a local minimum, or neither for

F_k

. Justify your answer using appropriate derivative tests. d. With the same value of

k

, identify all intervals on

(0,\pi)

where the graph of

F_k

is concave up and concave down, and locate any points of inflection. Support your conclusions with analytical evidence.