## zeesbrat3 one year ago Which one of the following statements is true?

1. zeesbrat3

Which one of the following statements is true? A.) If f is continuous on a closed interval [a, b], then f attains an absolute maximum value and f attains an absolute minimum inside the interval [a, b]. B.) If f ′′(c) = 0, then x = c is an inflection point on the graph of f(x). C.) If f ′′(x) < 0 on the interval (a, b), then f is concave up on the interval (a, b). D.) None are true.

2. zeesbrat3

3. UsukiDoll

well second derivative test does determine concavity

4. zeesbrat3

I remember there used to be a rule involving c, but I haven't studied this in 6 month so I am rusty. Can you explain?

5. UsukiDoll

haven't studied this in years, but still remember parts of it

6. Tommynaut

Tricky question. A) Can you think of any sort of curve or line that wouldn't have a maximum and minimum value in a domain? What is the definition of an absolute maximum/minimum? Does a horizontal line have an absolute maximum and minimum? B) What is always true when the second derivative at a point is 0? C) What is the difference between f''(x) > 0 and f''(x) < 0? Which one means concave up, which one means concave down?

7. UsukiDoll

we need the theorems/definitions

8. zeesbrat3

f''(x) < 0 i concave down

9. Tommynaut

Yep. It's a good question for you to look through your textbook or research online for, if you're unsure on definitions. This question is just testing your memory. And yes, 2nd deriv < 0 means down, > 0 means up. So C can't be true.

10. zeesbrat3

is* sorry. My keyboard i being temperamental

11. UsukiDoll

for all x in some interval f''(x) <0 is concave down and f''(x) >0 is concave up.

12. zeesbrat3

And on a closed interval, you don't necessarily have both a max and a min

13. UsukiDoll

this question is just matching theorems and definitions about graphing inflection, concavity, and continuity.. a calculus I topic. So at least we don't have to do that much. It's hard to type and eat pizza at the same time xD

14. zeesbrat3

15. Tommynaut

Can you give me an example where you wouldn't have a maximum and a minimum?

16. UsukiDoll

what about a graph that only has the absolute maximum?

17. zeesbrat3

Wait, does absolute max and min necessarily mean a real max/min? I'm not sure if I made sense..

18. UsukiDoll

hmm absolute max - highest point in the graph absolute min - lowest point in the graph relative max - a high point in the graph relative min - a low point in the graph

19. UsukiDoll

|dw:1436409136021:dw|

20. zeesbrat3

Ohh

21. Tommynaut

We're talking about a closed domain. In a closed domain, there would always be a highest and lowest value. The reason the question is tricky is because of option B. A lot of people think that f''(x) = 0 is enough to say a point is a point of inflection, but this isn't necessarily true.

22. UsukiDoll

if it's equal, f''(x) = 0 then it's neither a max or a min. It's just a constant

23. UsukiDoll

point of inflection is that particular area where the line changes from negative to positive. or vice versa

24. zeesbrat3

Oh! I mistook absolute min for a real min. Can yall help me with one more of this type?

25. Tommynaut

I mean, like in the example, if x=c and f''(c) = 0, this isn't enough information to say that there's a point of inflection at x=c.

26. Tommynaut

And a point of inflection is where the concavity of the line changes.

27. UsukiDoll

^ yeah...

28. zeesbrat3

Thank you. Ya, I read up on the point of inflections, but never really grasped it. It's weird..

29. zeesbrat3

Which one of the following statements is true? A.) If f ′′(x) > 0 on the interval (a, b) then f(x) is concave down on the interval (a, b). B.) If f ′(x) > 0 on the interval (a, b) then f(x) is increasing on the interval (a, b). C.) If f ′(c) = 0, then x = c is a relative maximum on the graph of f(x). D.) None of these are true.

30. Tommynaut

An example where f''(c) = 0 is not an inflection point is for the function y = x^4. It's like a fat parabola, so it obviously has no point of inflection (it's always concave up). However, f''(0) = 0, but x=0 is actually a minimum.

31. zeesbrat3

So A is wrong, because it is the opposite of what it should be

32. zeesbrat3

If there is a hip, there is a point?

33. Tommynaut

I'm not sure what you mean by that. But yes, you're right about A being wrong. B looks like it's right, so now look at C... does that look wrong to you? If f'(c) = 0, then sure, at x=c we MIGHT have a relative maximum, but it might be a minimum, or a saddle point.

34. zeesbrat3

I'm sorry, a saddle point? @Tommynaut

35. zeesbrat3

By hip, I meant a cubed function

36. Tommynaut

You might know a saddle point as a horizontal point of inflection (so f'(x) = 0 and f''(x) = 0).

37. zeesbrat3

Oh okay

38. Tommynaut

I'm off now, hope I helped

39. zeesbrat3

You did, thank you

40. UsukiDoll

I've heard of saddle, but it was for my Mathematical Biology class last semester. If a 2 x 2 matrix has one negative eigenvalue and one positive eigenvalue, then it's a saddle and that's always unstable. The graph looks like this |dw:1436410099737:dw|

41. UsukiDoll

$\det(A- \lambda I) = 0$ is what I used

42. zeesbrat3

Honestly, I have never seen those symbols before in my life

43. UsukiDoll
44. UsukiDoll

I think that saddle graph is a variation.. my book had what I drew

45. UsukiDoll