Mathematics

Problem

The function

g(x)=e^{kx}-mx^{2}

is defined for

x\ge 0

, where

k

and

m

are positive real parameters. (a) Compute

g''(x)

in terms of

k

and

m

. (b) Derive the inequality that

k

and

m

must satisfy so that

g

is concave down for every

x\ge 0

. (c) For parameter values that satisfy the inequality in part (b), determine whether

x=0

is a point of inflection of

g

. Justify your answer.