Homework 6

5. Let f be a convex function on the interval I and x 1 , x 2 and x 3 be points of I which
satisfy x 1 < x 2 < x 3 .
Prove that
f (x 2 ) − f (x 1 )
x 2 − x 1
≤ f (x 3) − f (x 1 )
x 3 − x 1
≤ f (x 3) − f (x 2 )
x 3 − x 2
.
6. Prove the following proposition (Jensen’s Inequality):
Suppose A is a convex set in R n and f : A → R is a concave function. Then, for any
integer m > 1,
( m
)
∑
m∑
f θ i x i ≥ θ i f ( x i)
i=1
i=1
∑
whenever x 1 , x 2 , ..., x m ∈ A, (θ 1 , θ 2 , ..., θ m ) ∈ R m + and m θ i = 1.
7. Let f : R + → R + be a strictly concave function. Let a, b, c, d be arbitrary positive
real numbers satisfying (i) a + d = b + c, and (ii) a < b < c < d. Show that
f (a) + f (d) < f (b) + f (c) .
i=1
8. If a and b are arbitrary positive real numbers, show that
a θ b 1−θ ≤ θa + (1 − θ) b, for every 0 < θ < 1.
9. Let f : R + → R + be a continuous function on R + which is twice continuously differentiable
on R ++ with f ′ (x) > 0 and f ′′ (x) < 0 for all x in R ++ . Show that for every
x in R ++ , f (x) > f ′ (x) .
x
10. Let f : R 2 → R be a quasi-concave function. Let I be the closed interval [0, 1] . Let
a, b be two arbitrary given vectors in R 2 . Define a function g : I → R by g (t) =
f (ta + (1 − t) b) . Prove that g is a quasi-concave function on I.
11. Suppose A is a convex set in R n and f : A → R. Prove that the following two statements
are equivalent to each other:
(a) f (x 2 ) ≥ f (x 1 ) implies f (θx 1 + (1 − θ) x 2 ) ≥ f (x 1 ) whenever x 1 , x 2 ∈ A, and
0 ≤ θ ≤ 1;
(b) For every α ∈ R, the set S (α) = {x ∈ A : f (x) ≥ α} is a convex set in R n .
3