Suppose that f: Rn †’ R is differentiable at a and that f(a) ‰  0.
a) Show that for ||h|| sufficiently small, f(a + h) ‰  0.
b) Prove that Df(a)(h)/||h|| is bounded for all h ˆˆ Rn{0}.
c) If T := -Df(a)/f2(a), show that

for ||h|| sufficiently small.
d) Prove that 1/f(x) is differentiable at x = a and

e) Prove that if f and g are real-valued vector functions which are differentiable at some a, and if g(a) ‰  0, then

Transcribed Image Text:

f(a + h)-f(a) _ T(h) = f(a)f (a + h) (fat. h)-f(a )) Df(a)(h) f3(a)/(a + h) D() (a)=-Dra) (a) =-T2(a) g(a) Df(a)-f(a) Dg(a) g2(a) a)