Show that f';(x) is continuous and f(x) has exactly two critical points, then f'(x) has a local maximum or local minimum between the two critical points.Help me With this true or false question?
FHelp me With this true or false question?
The first thing we need to think about is what it means for f(x) to have two critical points. A critical point is defined as a value of x for which f'(x) is 0. Thus, since there are two critical points, we know that f'(x) is zero at exactly two points.
Knowing this, we can apply Rolle's theorem, which states that if a function is continous on an interval [a,b], and if f(a) = f(b) then there is a point in that interval, lets call it c, such that f'(c) = 0.
In this case, our function if f'(x), and a and b are the two critical points. Since f' is zero at both these points, Rolle's theorem tells us that somewhere in between them is a critical point of f' -- where f'' is zero; hence, a maximum or minimum.
f no t i cant make it.
No comments:
Post a Comment