Mathematics

The key is to find the second derivative and determine its sign. Start by computing $y' = \frac{1}{1+x^2}$, then differentiate again to get $y''$. Since $x > 0$, you'll see that $y''$ is always negative—this means the curve bends downward everywhere on $(0,\infty)$, so the graph is concave down. Concavity tells you how the slope itself is changing: a negative second derivative means the function is increasing, but at a decreasing rate.

The key is to find the second derivative and determine its sign. Start by computing $y' = \frac{1}{1+x^2}$, then differentiate again to get $y''$. Since $x > 0$, you'll see that $y''$ is always negative—this means the curve bends downward everywhere on $(0,\infty)$, so the graph is concave down. Concavity tells you how the slope itself is changing: a negative second derivative means the function is increasing, but at a decreasing rate.