Mathematics

Problem

Fix a positive real number

p

and define

s_p(x)=\sin(px)+bx

on the closed interval

0\le x\le \pi

, where

b

is a real constant (which may depend on

p

). (a) Compute

s_p''(x)

in terms of

p

. (b) For the interval

0\le x\le \pi

, describe analytically the sub-intervals on which the sign of

s_p''(x)

depends only on

p

(that is, is independent of

b

). (c) Determine all positive values of

p

for which there exists at least one real

b

such that

s_p

is concave up on the entire interval

[0,\pi]

. Provide a rigorous justification and give an explicit description of the allowable

b

for those

p