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


f " > 0
f " < 0
f ' > 0
f ' < 0

so what is happening when f"<0 ?

To determine if the function's value will be greater than or equal to the local linear approximation, we need to consider the concavity (the second derivative) of the function.

The local linear approximation of a function f at a specific point is given by the equation:

L(x) = f(a) + f'(a)(x - a)

Here, f(a) is the value of the function at the point of tangency, f'(a) is the derivative of the function at that point, x represents any point in the interval, and a is the point where the tangent line is drawn.

If the second derivative, f''(x), is positive, it means that the function is concave up in that interval. In this case, the function's value will always be greater than or equal to the local linear approximation in the interval containing the point of tangency. Therefore, the correct answer is:

f'' > 0