If the graph of $y=a{x}^2+bx+c$ lies completely above the x-axis, then

Given that the graph of the equation $y=a{x}^2+bx+c$ lies above $x-$ axis

it implies that the graph will not intersect $x-$ axis, so it has no real roots.

 i.e., discriminant must be negative

$\Rightarrow b^2-4ac<0$

The graph has minimum value, then  $a>0$

The graph lies completely above the x-axis, If  $\boxed{\mathbf{\Delta }\mathbf{<}\mathbf{0}\mathbf{\text{ and a > 0}}}$