Calculus 1 Question concerning Concavity

Question

Give an example showing that a function can be concave up on its entire domain and yet never achieve a minimum value.

Answer 1

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

Answer 2

If the minimum value must be real:

f(x) = x² + 1