Math 445 and 447 - PRACTICE FINAL EXAM V1
Part D: Long Answer.
Math 445 students: Complete 7 problems from the following list.
Math 447 students: Complete 8 problems from the following list.
1. Show that a non-empty closed subset S of a compact metric space X is compact. HINT: Sequential
compactness is easier to deal with.
2. If d is a metric on X, show that d1(x, y) =
d(x,y)
1+d(x,y) is also a metric on X.
3. Suppose that W is a subspace of a normed linear space V . Show that the closure W is also a subspace.
4. Define a linear transformation T : C([0, 1])→ C([0, 1]) by
Tf(x) =
∫ x
0
f(t)g(t)dt
where g is a continuous function [0, 1]. Show that ‖T‖ ≤ ‖g‖∞.
5. Show that fn(x) = xne
−nx converges pointwise to zero but not uniformly on [0,∞).
6. Let F be a compact subset of C([0, 1]). Show that there is a function g ∈ F such that∫ 1
0
g(x)dx ≥
∫ 1
0
f(x)dx
for all f ∈ F . HINT: Show that the function sending f to ∫ 10 f is continuous in the appropriate context,
and therefore attains its maximum.
7. Show that
f(x, y) =
x
2y
√
x2+y2
x4+y2
(x, y) 6= (0, 0)
0 (x, y) = (0, 0)
is continuous at (0, 0) and that Dvf(0, 0) exists for all unit vectors v.
8. Show that f in problem 7 is not differentiable at (0, 0).
9. Suppose f : Rn → R is differentiable and f has a local minimum at x0. Show that Df(x0) = 0.
10. Consider the following non-linear equations:
x4 + y2 − 2z + w = 3
x− 2y + w4 = 0
Determine the points (x0, y0, z0, w0) for which the hypotheses of the implicit function theorem is satisfied.
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