If m is the slope of the tangent to the curve e^y=1+x^2, then

We have,

$e^y=1+x^2$

$\Rightarrow e^y \frac{d y}{d x}=2 x$             [Differentially w.r.t. x]

$\Rightarrow \left(1+x^2\right) \frac{d y}{d x}=2 x$               [∵ $e^y=1+x^2$]

$\Rightarrow \frac{d y}{d x}=\frac{2 x}{1+x^2}$

$\Rightarrow |m|=\frac{2|x|}{1+|x|^2}$

Now, A.M. $\geq$ G.M.

$\Rightarrow \frac{1+|x|^2}{2} \geq \sqrt{1 \times|x|^2}$

$\Rightarrow \frac{1+|x|^2}{2} \geq|x|$

$\Rightarrow 1+|x|^2 \geq 2|x|$

$\Rightarrow 1 \geq \frac{2|x|}{1+|x|^2} \Rightarrow 1 \geq|m| \Rightarrow|m| \leq 1$