math analysis

Midterm 2

Due

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1.13.13 by 3pm. Submit the exam to me in office 1119 of
WWH. Late exams will not be accepted.

1.(20 points) Let f : Rd → R be a C1 function such that

f(ax) = akf(x)

for any a ∈ R where k ≥ 1 is an integer. Show that

∇f(x) ·x = kf(x)

2.(10 points) Let f(x,y) = cos(ex + 3y). Compute D2f.

Remark: D2f is just the derivative of the ∇f.

3.(20 points) Let Ω ⊂ Rd be open. Suppose that f : Ω → R satisfies

d∑
j=1

∂2f

∂x2j
= 0.

Let φ : R → R be a C∞ function and assume it is convex (also known as
concave up). Show that g(x) = φ(f(x)) satisfies

d∑
j=1

∂2g

∂x2j
≥ 0,

when x ∈ Ω.

4.(20 points) Use Taylor’s theorem to prove the expansion

(x + y)n =
n∑

k=0

(
n
k

)
xn−kyk

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