Given,

\[f\text{ }\left( x \right)\text{ }=\text{ }2x\text{ }+\text{ }bed\text{ }1\text{ }x\text{ }+\text{ }log\text{ }\left[ \surd \left( 1\text{ }+\text{ }x2 \right)\text{ }\text{ }x \right]\]

differentiating the two sides w.r.t. x, we get

On figuring out both the sides, we get

\[4×4\text{ }+\text{ }1\text{ }+\text{ }4×2\text{ }\ge \text{ }1\text{ }+\text{ }x2\]

\[4×4\text{ }+\text{ }4×2\text{ }\text{ }x2\text{ }\ge \text{ }0\]

\[4×4\text{ }+\text{ }3×2\text{ }\ge \text{ }0\]

\[x2\left( 4×2\text{ }+\text{ }3 \right)\text{ }\ge \text{ }0\]

The above is valid for any worth of x ∈ R.

In this manner, the given capacity is an expanding capacity over R.