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

Question

If f(x)=(x2-1/x2+1) ∀ x ∈ R then the minimum value of f (A) does not exists because it is unbounded (B) is equal to 2 (C) is cqual to 5 (D) is cqual to -1

Tardigrade Admin · Accepted Answer

f(x)=1-(2/x2+1) Now x2+1 ≥ 1 ⇒ (1/x2+1) ≤ 1 ⇒-(2/x2+1) ≥-2 ⇒ f(x) ≥-1

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

does not exists because it is unbounded

is equal to 2

is cqual to 5

is cqual to $- 1$

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

Q. If f(x)=x2+1x2−1​∀x∈R then the minimum value of f