17. Define http://answerboard.cramster.com/Answ...9089902240.gif by f(x) = 1/(1 + x2). Prove that f has a maximum and find the point at which that maximum occurs.

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- Apr 27th 2008, 06:25 AMlacy1104maxium at a point
17. Define http://answerboard.cramster.com/Answ...9089902240.gif by f(x) = 1/(1 + x2). Prove that f has a maximum and find the point at which that maximum occurs.

Thanks - Apr 27th 2008, 06:37 AMIsomorphism
Since $\displaystyle \forall x \in \mathbb{R}, x^2 \geq 0 \text{ Equality at x=0}$

$\displaystyle \Rightarrow 1+x^2 \geq 1 \text{ Equality at x=0}$

$\displaystyle \Rightarrow \frac1{1+x^2} \leq 1 \text{ Equality at x=0}$

Thus f(x) is bounded above by 1(its also the lub and hence the maxima).

We also see that maxima occurs at 0.