What property must f satisfy if the local linear approximation of f is always greater than or equal to the function's value for all x in an interval containing the point of tangency?

d^2f/dx^2 <0

in other words curves down, sheds water

To determine the property that f must satisfy for its local linear approximation to be greater than or equal to the function's value for all x in an interval containing the point of tangency, we need to consider the conditions for the local linear approximation.

The local linear approximation of a function f at a point (a, f(a)) is given by the equation of the tangent line to the graph of f at that point. This tangent line has the general form:

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

Where f'(a) represents the derivative of f evaluated at x = a.

In order for the local linear approximation to be greater than or equal to the function's value for all x in an interval, we need to ensure that the tangent line lies above or touches the graph of f in that interval.

If the tangent line is always above or touching the graph of f, it means the function is concave up in that interval. This condition can be described mathematically by saying that the second derivative of f is non-negative on the given interval.

Therefore, the property that f must satisfy is that its second derivative is non-negative on the interval that contains the point of tangency. In other words, f''(x) ≥ 0 for all x in the interval.