Question

The local linear approximation of a function f will always be greater than or equal to the function's value if, for all x in an interval containing the point of tangency,

f '(x) < 0
f '(x) > 0
f "(x) < 0
f "(x) > 0

Answer 1

@HolsterEmission

Answer 2

Consider \(f(x)=x^2\) and the point \((0,0)\) as the point of tangency.

The tangent line to this is just the line \(y=0\). It's true that \(f(x)\ge y\) for all \(x\), but to one side of the line \(x=0\), you have a negative derivative, while to the other it's positive.

This example shows that the inequality in question has nothing to do with how the first derivative behaves.