1 Concavity and Monotonicity (Theory)
In this problem, we will see the interplay between concavity, monotonicity and maxima. Different
functions are given below; you will need to prove and/or disprove the concavity of each and find
the maximizer and maximum value. For each part, make sure to highlight your final answer
or conclusion by changing the font colour to red or drawing a box around it. Number all
equations that you refer to. Make sure to carefully follow the submission instructions on the
first page of the assignment.
(a) Given a function:
f (x) = −x2 + 2x − 1 (1)
Prove that the function f (x) is strictly concave.
(b) Prove that f (x) has a unique global maximum. Find the global maximizer and the global
maximum value, i.e.: arg maxx f (x) and maxx f (x).
(c) Now consider a new function
g(x) = e(−x2+2x−1) (2)
Prove that g(x) is not a concave function.
(d) Consider two functions ψ : R → R and ϕ : R → R, which are not necessarily differentiable.
ψ(·) is strictly concave, and ϕ(·) is concave and strictly increasing.
Now, consider the composition of these functions:
h(x) = (ϕ ◦ ψ)(x) = ϕ(ψ(x)) (3)
Prove that the function h(x) is also a strictly concave function.
Note: Because ψ(x) and ϕ(x) may not be differentiable, neither the first-order or the second
order derivatives of ψ(x) and ϕ(x) are necessarily well-defined and therefore should not be used
in your proof.
(e) Consider the same definitions of ψ and ϕ as in the previous part, with one difference: now
ϕ(·) is strictly increasing, but not necessarily concave. ψ(·) is strictly concave is still strictly
We again consider the composition of these functions:
c(x) = (ϕ ◦ ψ)(x) (4)
Is c(x) concave? If c(x) is concave, prove its concavity. If c(x) is not a concave function,
give a counterexample, that is, show an example of a function c(x) that satisfies all the
above conditions on ψ and ϕ, but is not concave.
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