Mathematics

Problem

Consider the family of functions

g_m(x)=mx^3+2x-1

, where

m

is a real parameter. Answer the following questions with respect to the interval

[0,1]

. a. Express

g_m(0)

and

g_m(1)

in terms of

m

. Briefly interpret the sign of each expression. b. Determine all real values of

m

for which the equation

g_m(x)=0

has at least one root in

(0,1)

. Provide complete justification. c. For one specific value of

m

found in part (b), choose a non-trivial subinterval of

[0,1]

of length at most

\tfrac14

that must contain a root. Show your reasoning.