There are several equivalent definitions of concavity. We say a twice-differentiable function ƒ to be (strictly) concave up on (a, b) if for any distinct x₁, x2 € (a, b) (assume x1 < x2) and for any t € (0, 1) we have f\left(x_{1}+t\left(x_{2}-x_{1}\right)\right)