Solution: Convex: is convex if domain of f, is a convex set
and .
is strictly convex if domain of f, is a convex set
and
(h) Let is convex and is strictly convex and non decreasing.
Composition Rules:(i) is convex if
f is convex; h convex non-decreasing
(ii) is strictly convex if
f is convex; h strictly convex non-decreasing.
Since h is non-decreasing, we have . Since h is convex, we have .
Since f is convex, we have .
Therefore,
and
Since f is convex and h strictly convex and non-decreasing'
Therefore is strictly convex.
Thus the statement (h) is true.
(i) A continuous function f(x) that is defined on all of is coercive if
.
That is for any constant M>0 there exists a constant such that
whenever .
Therefore if f is strictly convex, then it need not be coercive.
So statement is false.
(j) Let is such that the level set
for every .
If is convex then for every is convex but converse is false that is if is such that the level set
is convex for every then need not
be convex. Therefore statement is false.
Epigraph of :
f is convex if and only if epi(f) is a convex set.
(k) If f is convex and coercive, then it is strictly convex.
So statement is true.
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