In Questions 1-5 prove your answer. In this test you can use
without proofs theorems that were proved in the lectures or in the
book, just give a reference.
1.(10 pts) Let f be a function on [−1, 1]. Two of the following state-
ments, if combined, imply that f′(0) = 0. Which two statements?
A: limx→0 f(x) = 0;
B:Thesequence{n(f(n1)−f(0))}∞n=1 haslimitzero;
C: f is differentiable at zero;
D: f is uniformly continuous on [−1, 1].
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2. (10 pts) Suppose that f, g are continuous functions on [0, 1], dif-
ferentiable on (0, 1), g′(x) ̸= 0 for all x ∈ (0, 1), and
f(0)=1, f(1)=2, g(0)=3, g(1)=1.
It is known that three of the following statements are true, while one
is not. Which one is not true? Explain your answer.
A: f − g is an increasing function on (0, 1);
B: f + g is a decreasing function on (0, 1);
C: f′(x)g′(x) > 0 for every x ∈ (0,1);
D: f′(x)g′(x) < 0 for every x ∈ (0,1);
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3. (10 pts) Which of the following polynomials are increasing in a
neighborhood of zero? Prove your answer.
A: 100×7 + 2×3 + x2
B: 5×14 −7×7 −x5
C: x10 +2×6 +4×3
D: 4×9 −3×8 −5×6
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4.(10 pts) Which of the following inequalities are correct? Prove your
answer. You are not allowed to use a calculator for this problem.
A: 23/2 + 63/2 + 73/2 ≥ 3 · 53/2;
B: 2−3/2 + 6−3/2 + 7−3/2 < 3 · 5−3/2;
C:e2x +e6x +e7x ≥3e5x foreveryx∈R;
D:e−2x +e−6x +e−7x <3e−5x foreveryx∈R.
Hint: If f is a convex function then
x1 +…+xn f(x1)+…+f(xn)
fn≤n.
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5. (10 pts) Let f be a 1-1 continuous function on [0, 2]. Which three
of the following statements combined imply that f is differentiable at
1 and f′(1) = 81? Prove your answer.
A:f(0)=3, f(2)=6;
B: f(1) = 8;
C: f−1 is differentiable at 5 and (f−1)′(5) = 8;
D: f(1) = 5.
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6. (10 pts) (i) Suppose that f is a differentiable function everywhere
on R, and f′(x) = 3 for every x ∈ R. Prove that f(x) = 3x+b, where
b is a constant.
(ii) Suppose that f is a differentiable positive function on R such
that
f′(x) = 3f(x)
for every x ∈ R. Prove that
f(x) = Ce3x,
where C > 0 is an arbitrary positive constant.
7. (10 pts). Prove that for every polynomial
P(x)=a0 +a1x+a2x2 +….+anxn, an ̸=0
there exists M ∈ R so that P is strictly increasing or strictly decreasing
on [M, ∞).
Hint: Use a theorem from algebra that a polynomial of degree k has
no more than k roots, and prove that P is 1-1 on some interval [M, ∞).
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8. (10 pts) Suppose that f is twice continuously differentiable in a
neighborhood of the point a, and f′′(a) = 3. Compute
f(x) − f(a) − f′(a)(x − a)
lim 2 .
x→a (x − a)
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9. (10 pts) Suppose that p > 0, and the function f is equal to
|x|p cos( x1 ) if x ̸= 0, and f (0) = 0. Prove that f is differentiable at zero
if and only if p > 1.
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10. (10 pts) Prove that for every x > 0
1 + 2 ln x ≤ x .
2
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Extra Credit (10 pts). Suppose that f is twice continuously differ-
entiable on [0,1], f(0) = f(1) = 0 and |f′′(x)| ≤ A for every x ∈ (0,1).
Prove that |f′(x)| ≤ A2 for every x ∈ [0,1].
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