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
a. f’’>0
B. F’’<0
C.f’>0
D.f’<0
To determine which option satisfies the given condition, we need to understand the concept of local linear approximation and the relationship between the function's value and its second derivative.
The local linear approximation of a function f at a point a is given by the equation of the tangent line to the graph of f at that point. The tangent line is the best linear approximation of f near the point of tangency.
The condition states that the function's value will always be greater than or equal to the local linear approximation of f if, for all x in an interval containing the point of tangency.
For the function f, to have a local minimum (where the function's value is greater than or equal to the local linear approximation), it must satisfy the condition that the second derivative is positive (f'' > 0). This is because a positive second derivative indicates a concave-up graph, where the tangent line lies below the curve.
Therefore, the correct answer is (A) f'' > 0.
sketch graph
increasing faster, curving above line,
a.