Use the Mean Value Theorem to prove that sqrt(1+x) < 1 + x/2 for all x > 0. Generalize this result to the function (1+x)^r, where r < 1 is a positive rational number.
Use the Mean Value Theorem to prove that sqrt(1+x) < 1 + x/2 for all x > 0. Generalize this result to the function (1+x)^r, where r < 1 is a positive rational number.
Thanks!
Using the MVT, for some .
. For , .
Thus,
for
as desired.
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Using the same method, see if you can generalize that result.