Please help, T/F1. _____ If f'(x) =0 when x=c then f has either a minimum or maximum at x=c.
2. _____ If a differentiable function f has a minimum or maximum at x=c, then f'(c)=0.
3. _____ If f is continuous on an open interval (a, b) then the f attains maximum or minimum in (a, b).
4. _____ If f'(x)=g'(x) then f(x) = g(x).
5. _____ If x=c is an inflection point for f, then f (c) must be a maximum or minimum of f.
6. _____ f(x) = ax2 +bx +c, (with a ≠ 0), can have only one critical point.
7. _____ Second Shape Theorem includes the converse of First Shape Theorem.
8. _____ If f(x) has a minimum at x=a, then there exists an ε, such that f(x) > f(a) for every x in (a- ε, a+ ε).
9. _____ The mean value theorem applies as long as the function is continuous on an interval [a, b].
10. _____ If f (x) has an extreme value at x=a then f is differentiable at x=a.
11. ______If f(x) is continuous everywhere, and f(a)=f(b), then there exists x=c such that f'(c)=0.
12. ______ if f(x) is continuous and differentiable everywhere, then f attains a max or min at x=a, if f'(a)=0.
13. ______The function f (x) =x3 does not have an extreme value over the closed interval [a, b].
14. ______If f(x) is not differentiable at x=a, then (a, f(a)) cannot be an extreme value of f.
15. ______If f"(a-ε)*f"(a+ε) <0, for an arbitray positive number ε, then the function f(x) has an extreme value at x=a.
16. ______If f'(a-ε)*f'(a+ε) <0, for an arbitray positive number ε, then the function f(x) has an extreme value at x=a whenever f'(a)=0, or f'(a) is undefined.
17. ______The function y=(x-2)3 + 1 has an inflection point at (2, 1)
18. ______The function: is always increasing in (0, + ).
19. ______If f'(a)=0, and f"(a)>0 then f has a minimum at x=a.
20. ______If f"(x)>0 on an interval I then f(x) is increasing on I.