Let $f:R\to \mathrm{R}$  be a differentiable function $\mathrm{\forall }x\in R$.  If the tangent drawn to the curve at any point  $x\in (a,b)$always lies below the curve, then

$f^{\mathrm{\prime }}\left(x\right)<0,f^{\prime \prime }\left(x\right)<0$

Clearly for $f\text{'}\left(x\right)>0,f^{\prime \prime }\left(x\right)>0\text{[i.e figure}\left(\mathrm{c}\right)]\text{,}$ tangent always lies below the graph. 

for $f^{\mathrm{\prime }}\left(x\right)<0,f^{\prime \prime }\left(x\right)>0[\text{i.e figure (d) ],}$ tangent always lies below the graph.