Let f be a real-valued function defined on [0,∞), with the properties: f is
continuous on [0, ∞), f(0) = 0, f
0
exists on (0, ∞), and f
0
is monotone increasing
on (0,∞).
Let g be the function given by: g(x) = f(x)
x
for x ∈ (0, ∞).
a) Prove that g is monotone increasing on (0,∞).
b) Prove that, if f
0
(c) = 0 for some c 0, and if f(x) ≥ 0, for all x ≥ 0,
then f(x) = 0 on the interval [0, c].

Question

Let f be a real-valued function defined on [0,∞), with the properties: f is
continuous on [0, ∞), f(0) = 0, f
0
exists on (0, ∞), and f
0
is monotone increasing
on (0,∞).
Let g be the function given by: g(x) = f(x)
x
for x ∈ (0, ∞).
a) Prove that g is monotone increasing on (0,∞).
b) Prove that, if f
0
(c) = 0 for some c > 0, and if f(x) ≥ 0, for all x ≥ 0,
then f(x) = 0 on the interval [0, c].

confluxepic · Answer

Wrong section. @Darksteel @~