True or False (Optional Explanation)

1._____ Second Shape Theorem includes the converse of First Shape Theorem.

2._____ If f(x) has a minimum at x=a, then there exists an , such that f(x) > f(a) for every x in (a- , a+ ).

3._____ The mean value theorem applies as long as the function is continuous on an interval [a, b].

4._____ If f (x) has an extreme value at x=a then f is differentiable at x=a.

5.______If f(x) is continuous everywhere, and f(a)=f(b), then there exists x=c such that f'(c)=0.

6.______ if f(x) is continuous and differentiable everywhere, then f attains a max or min at x=a, if f'(a)=0.

7.______The function f (x) =x³ does not have an extreme value over the closed interval [a, b].

8.______If f(x) is not differentiable at x=a, then (a, f(a)) cannot be an extreme value of f.

9.______If f"(a-)*f"(a+) <0, for an arbitray positive number , then the function f(x) has an extreme value at x=a.

10.______If f'(a-)*f'(a+) <0, for an arbitray positive number , then the function f(x) has an extreme value at x=a whenever f'(a)=0, or f'(a) is undefined.

11.______The function y=(x-2)³ + 1 has an inflection point at (2, 1)

12.______The function: f(x)=(1+x)^3/2 -(3/2x)-1 is always increasing in (0, + ).

13.______If f'(a)=0, and f"(a)>0 then f has a minimum at x=a.

14.______If f"(x)>0 on an interval I then f(x) is increasing on I.