Math 4100/6100 Addition Problems 5
1.* Suppose c∈R,fis a twice differentiable real-valued function on (c, ∞),
and M0, M1, M2are least upper bounds for |f(x)|,|f0(x)|,|f00 (x)|re-
spectively, on (c, ∞). Prove that M2
1≤4M0M2.
2.* A real-valued function is said to be convex if
f(λx + (1 −λ)y)≤λf(x) + (1 −λ)f(y)
whenever a<x<b,a < y < b, and 0 < λ < 1.
(a) Prove that if fis differentiable, then fis convex if and only if f0
is increasing.
(b) Assume next that f00(x) exists for every x∈(a, b), and prove that
fis convex if and only if f00(x)≥0 for all x∈(a, b).
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