Please help and thank you!

Question

The local linear approximation of a function f will always be less than or equal to the function's value if, for all x in an interval containing the point of tangency,

Options:

1. f ' < 0
2. f ' > 0
3. f " < 0
4. f " > 0

Answer 1

The local linear approximation will be less than or equal to a functions value if f '' > 0, (choice 4).

Remember that f '(x) is the slope of f (x).

f '' > 0 (also known as concave up) means that f ' is increasing (because f '' is the derivative, or rate of change, of f ' ). The local linear approximation takes the value of f ' at a point and assumes that close to that point, the slope stays constant. Since in fact f '(x), the slope, is increasing, assuming that it is constant will lead to an approximation that is less than the actual value.

Hope this helps clear up the concepts behind what f '' tells you about a function!