what is the Behrouli inequelity
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In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of 1 + x. It is often employed in real analysis.
An illustration of Bernoulli's inequality, with the graphs of {\displaystyle y=(1+x)^{r}}{\displaystyle y=(1+x)^{r}} and {\displaystyle y=1+rx}{\displaystyle y=1+rx} shown in red and blue respectively. Here, {\displaystyle r=3.}r=3.
The inequality states that
{\displaystyle (1+x)^{r}\geq 1+rx}{\displaystyle (1+x)^{r}\geq 1+rx}
for every integer r ≥ 0 and every real number x ≥ −1.[1] If the exponent r is even, then the inequality is valid for all real numbers x. The strict version of the inequality reads
{\displaystyle (1+x)^{r}>1+rx}{\displaystyle (1+x)^{r}>1+rx}
for every integer r ≥ 2 and every real number x ≥ −1 with x ≠ 0.
There is also a generalized version that says for every real number r ≥ 1 and real number x ≥ −1,
{\displaystyle (1+x)^{r}\geq 1+rx,}{\displaystyle (1+x)^{r}\geq 1+rx,}
while for 0 ≤ r ≤ 1 and real number x ≥ −1,
{\displaystyle (1+x)^{r}\leq 1+rx.}{\displaystyle (1+x)^{r}\leq 1+rx.}
Bernoulli's inequality is often used as the crucial step in the proof of other inequalities. It can itself be proved using mathematical induction, as shown below.
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