232 CHAPTER 8. IMPLICIT FUNCTION THEOREM

10. Suppose B is an open ball in X and f : B→ Y is differentiable. Suppose also thereexists L∈L (X ,Y ) such that ∥Df (x)−L∥< k for allx∈B. Show that ifx1,x2 ∈B,

|f (x1)−f (x2)−L(x1−x2)| ≤ k |x1−x2| .

Hint: Consider Tx= f (x)−Lx and argue ∥DT (x)∥< k.

11. Let f : U ⊆ X → Y , Df (x) exists for all x ∈ U , B(x0,δ ) ⊆ U , and there existsL ∈L (X ,Y ), such that L−1 ∈L (Y,X), and for all x ∈ B(x0,δ )

∥Df (x)−L∥< r∥L−1∥ , r < 1.

Show that there exists ε > 0 and an open subset of B(x0,δ ) called V , such thatf : V → B(f (x0) ,ε) is one to one and onto. Also Df−1 (y) exists for each y ∈B( f (x0) ,ε) and is given by the formula Df−1 (y) =

[Df(f−1 (y)

)]−1. Hint: Let

Ty (x)≡ T (x,y)≡ x−L−1 (f (x)−y)

for |y−f (x0)|< (1−r)δ2∥L−1∥ , consider {T n

y (x0)}. This is a version of the inverse func-

tion theorem for f only differentiable, not C1.

12. If f is one to one and C1, and Df (x0) is invertible, then locally the function f isone to one. Explain why this is, maybe using the above problem. However, this is astrictly local result! Let f : R2→ R2 be given by

f (x,y) =(

ex cosyex siny

)This clearly is not one to one because if you replace y with y+2π, you get the samevalue. Now verify that Df (x,y)−1 exists for all (x,y).

13. Show every polynomial, ∑|α|≤k dαxα is Ck for every k. Show that if f is definedand continuous on a compact set K, then there is an infinitely differentiable functionwhich is uniformly close to f on K.

14. Suppose U ⊆ R2 is an open set and f : U → R3 is C1. Suppose Df (s0, t0) has ranktwo and

f (s0, t0) =(

x0 y0 z0)T

.

Show that for (s, t) near (s0, t0), the points f (s, t) may be realized in one of thefollowing forms.

{(x,y,φ (x,y)) : (x,y) near (x0,y0)},

{(φ (y,z) ,y,z) : (y,z) near (y0,z0)},

or{(x,φ (x,z) ,z,) : (x,z) near (x0,z0)}.

This shows that parametrically defined surfaces can be obtained locally in a particu-larly simple form.

15. Minimize ∑nj=1 x j subject to the constraint ∑

nj=1 x2

j = a2. Your answer should be somefunction of a which you may assume is a positive number.