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Is $\mathrm{cos}\left(x\right)\ge 1-\frac{{x}^{2}}{2}$ for all x in R?
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jugf5
Consider the function
$f\left(x\right)=\mathrm{cos}x-1+\frac{{x}^{2}}{2}$
We have f(0)=0 and
${f}^{\prime }\left(x\right)=x-\mathrm{sin}x$
with f′(0)=0. Also
${f}^{″}\left(x\right)=1-\mathrm{cos}x$
which shows that f′(x) is a strictly increasing function, because its derivative is positive except on a set of isolated points (that has no limit point). Therefore f′(x)>0 for x>0 and f′(x)<0 for x<0. Hence 0 is an absolute minimum for f. Since f(0)=0, we have f(x)>0 for every $x\ne 0$. This means that, for every x,
$\mathrm{cos}x\ge 1-\frac{{x}^{2}}{2}$
equality holding only for x=0.