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
To determine whether the local linear approximation of a function f will always be greater than or equal to the function's value, we need to look at the sign of the derivative of the function f at each point in an interval containing the point of tangency.
The local linear approximation of a function f at a point x is given by the equation of the tangent line to the graph of f at that point. This tangent line can be expressed as:
y = f(a) + f'(a)(x - a)
where a is the point of tangency and f'(a) is the derivative of f at that point.
Now, if we want the local linear approximation to always be greater than or equal to the function's value, the function must be increasing or at least not decreasing in the interval containing the point of tangency. This means that the derivative f'(x) should be greater than or equal to zero for all x in the interval.
Therefore, the correct answer is:
f'(x) ≥ 0