Fernando G. answered • 01/05/20

Notre Dame Grad specializing in chemistry, physics, math, theology

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!