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2017 IMC
2
IMC 2017 Problem 2
IMC 2017 Problem 2
Source:
August 2, 2017
function
IMC
imc 2017
calculus
Problem Statement
Let
f
:
R
→
(
0
,
∞
)
f:\mathbb R\to(0,\infty)
f
:
R
→
(
0
,
∞
)
be a differentiabe function, and suppose that there exists a constant
L
>
0
L>0
L
>
0
such that
∣
f
′
(
x
)
−
f
′
(
y
)
∣
≤
L
∣
x
−
y
∣
|f'(x)-f'(y)|\leq L|x-y|
∣
f
′
(
x
)
−
f
′
(
y
)
∣
≤
L
∣
x
−
y
∣
for all
x
,
y
x,y
x
,
y
. Prove that
(
f
′
(
x
)
)
2
<
2
L
f
(
x
)
(f'(x))^2<2Lf(x)
(
f
′
(
x
)
)
2
<
2
L
f
(
x
)
holds for all
x
x
x
.
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