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DTSC 615: Optimization Methods for Data
Science
Homework #01 Solutions
1) (5 ×4 = 20 points) Determine if each of the function depicted in the graphs shown in Fig. 1 is convex,
concave, neither or both. Give reasons for your answer.
[Remark: No points for just writing convex or concave or both or neither.]
(a)[Solution: Neither. See the red line in the graph (a) in Fig. 2. It is partly above the graph and partly
below.]
(b)[Solution: Concave: See the red line in the graph (b) in Fig. 2. It is always below the graph.]
(c)[Solution: Convex: See the red line in the graph (c) in Fig. 2. It is always above the graph.]
(d)[Solution: Concave: See the red line in the graph (d) in Fig. 2. It is always below the graph or sometimes
on the graph.]
(e)[Solution: Both: The function is a straight line.]
x
f(x)
x
f(x)
x
f(x)
x
f(x)
x
f(x)
(a) (b)
(d)
(c)
(e)
Fig. 1. Figures for Problem # 1).
x"
f(x)"
x"
f(x)"
x"
f(x)"
x"
f(x)"
x"
f(x)"
(a)" (b)"
(d)"
(c)"
(e)"
Fig. 2. Solutions for Problem # 1).
