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
The local linear approximation of a function f at a point of tangency is given by the equation of the tangent line to the graph of f at that point. To determine whether the approximation will always 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 behavior of the function and its derivatives.
The first derivative of a function, f'(x), represents the rate of change of the function at a given point. If f '(x) is negative, it means that the function is decreasing in value at that point. In this case, the local linear approximation of f will be greater than the function's value because the tangent line, being a line with negative slope, will be positioned above the graph of f.
Conversely, if f '(x) is positive, it means that the function is increasing in value at that point. In this scenario, the local linear approximation of f will be less than the function's value because the tangent line, having a positive slope, will be positioned below the graph of f.
Therefore, the correct answer is: f '(x) > 0. If the first derivative is greater than zero, the local linear approximation will always be less than or equal to the function's value over the given interval.