Question

The largest open interval in which the function f(x) = $\frac{1}{1+x^2}$ decreases is _______________.

Answer 1

The given function is $f\left(x\right)=\frac{1}{1+x^2}$.

$f\left(x\right)=\frac{1}{1+x^2}$

Differentiating both sides with respect to x, we get

$f\text{'}\left(x\right)=-\frac{2x}{{\left(1+x^2\right)}^2}$

For f(x) to be decreasing,

$f\text{'}\left(x\right)<0$

$\Rightarrow -\frac{2x}{{\left(1+x^2\right)}^2}<0$

$\Rightarrow -2x<0\left[{\left(1+x^2\right)}^2>0\forall x\in \mathrm{R}\right]$

$\Rightarrow x>0$

∴ x ∈ (0, ∞)

Thus, the largest open interval in which the given function f(x) decreases is (0, ∞).


The largest open interval in which the function f(x) = $\frac{1}{1+x^2}$ decreases is ____(0, ∞)____.