Question

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,

Answer 1

f'' > 0

A positive second derivative implies that the function is concave up in the interval, which means it's greater or equal to the tangent line at any point in the interval.

Answer 2

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 (first option)

Explanation:

When the second derivative f'' is greater than zero (f''>0), the graph of the function f is concave up, and all x in an interval containing the point of tangency are above the tangent line in the point of tangency; that means 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.