Exercise 5.3.6. (a) Let

g:[0,a]->R

be differentiable,

g(0)=0

, and

|g^(')(x)|

for all

xin[0,a]

. Show

|g(x)|

for all

xin[0,a]

.\ (b) Let

h:[0,a]->R

be twice differentiable,

h^(')(0)=h(0)=0

and

|h^('')(x)|

M

for all

xin[0,a]

. Show

|h(x)|

for all

xin[0,a]

.\ (c) Conjecture and prove an analogous result for a function that is differentiable three times on

0,a

.

g(x)M for all x[0,a]. Show g(x)Mx for all x[0,a]. (b) Let h:[0,a]R be twice differentiable, h(0)=h(0)=0 and h(x) M for all x[0,a]. Show h(x)Mx2/2 for all x[0,a]. (c) Conjecture and prove an analogous result for a function that is differentiable three times on [0,a]