f(x) = e^x
f'(x) = e^x
f"(x) = e^x > 0 for all x, the slope is always increasing and positive, f(x) is U shaped, at least the right half of a U, but never the left half or the bottom of a U.
concave upwards means the slope is everywhere increasing, as with f(x) = e^x
It's concave up for all x, yet never achieves a minimum. It only approaches zero as a minimum but never reaches it. f(x)>0 for all x but never = 0
Other exponential functions have the same characteristic, concave up everywhere, but never achieve a minimum value.
a hyperbolic function xy=a constant or y= f(x) = c/x at least in Quadrant I is concave upwards everywhere, yet also never achieves zero as a minimum value. It just approaches zero as a limit.
f(x) = 1/x^2 is concave upwards on all values of x where x = any real number except zero
and 1/x^2 has no minimum, just a limit of zero as x approaches either infinity or negative infinity