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  4. If f(x)=(x2-1/x2+1) ∀ x ∈ R then the minimum value of f

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Q. If $f(x)=\frac{x^2-1}{x^2+1} \forall x \in R$ then the minimum value of $f$

Complex Numbers and Quadratic Equations

Solution:

$f(x)=1-\frac{2}{x^2+1}$
Now $x^2+1 \geq 1 $
$\Rightarrow \frac{1}{x^2+1} \leq 1$
$ \Rightarrow-\frac{2}{x^2+1} \geq-2$
$ \Rightarrow f(x) \geq-1$