Proving Inequalities using the MVT
Wednesday, 28 March 2018
2:27 PM
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Proving inequalities using the MVT
An important application of the M VT is in proving inequalities.
Example. Show that
> for all .r > 0.
Hint: Fix x > 0 and apply the MVT with [a, b] [0, x] for this fixed x > 0.
Solution. Suppose that x > O and consider the closed interval [0, x]. We define
t
the function f : [0, x] + R by f (t) —
Now, f is continuous on [0, x] and differentiable on (0, x) so that MVT implies
that there exists c (0, x) such that
f(x) — f(0)
f'(c)
A lower bound of f' ( )
c on (0, x) is given by f' (c)
C
ex—I
We therefore conclude that
> 1 or, equivalently
Alina Ostafe (UNSW)
Chapter 5: The Mean Value Theorem
—1
1.
Semester 1 2018
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