Let

f:[a,b]->R

be continuous on

a,b

and differentiable on

(a,b)

.\ (a). Prove that if

f^(')(x)!=0

in

(a,b)

then

f

is injective (one-to-one).\ (b). Give an example of a function

f:R->R

that is one-to-one, and such that

f^(')(x_(0))=0

for some

x_(0)inR

.\ (c). Prove that if

f^(')(x)>0

in

(a,b)

, then

f

is strictly increasing in

a,b

.\ (d). Prove that if

f^(')(x)

in

(a,b)

, then

f

is strictly decreasing in

a,b

.

1. Let f:[a,b]R be continuous on [a,b] and differentiable on (a,b). (a). Prove that if f(x)=0 in (a,b) then f is injective (one-to-one). (b). Give an example of a function f:RR that is one-to-one, and such that f(x0)=0 for some x0R. (c). Prove that if f(x)>0 in (a,b), then f is strictly increasing in [a,b]. (d). Prove that if f(x)0 in (a,b), then f is strictly increasing in [a,b]. (d). Prove that if f(x)