Question

Does y= x^2 have concave upwards on the left and right of the critical point?

Answer 1

Concavity is determined by the second derivative.
The critical point is determined by the solution of the first derivative = zero. The convention is that concave up means the second derivative is positive. So the answer is ...

Answer 2

Yeah it's concave upwards because f''(x) is positive, but I was wondering if y = x^2 would fit the following description:

f'(x) < 0 and f''(x) > 0 at (-infinity, 0)
f'(x) > 0 and f''(x) > 0 at (0, infinity)

Answer 3

That is true. The first derivative changes sign at the origin, but the second derivative is always positive.

Answer 4

Ok thank you!