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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.
well second derivative test does determine concavity
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?
haven't studied this in years, but still remember parts of it
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?
we need the theorems/definitions
f''(x) < 0 i concave down
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.
is* sorry. My keyboard i being temperamental
for all x in some interval f''(x) <0 is concave down and f''(x) >0 is concave up.
And on a closed interval, you don't necessarily have both a max and a min
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
Hahaha I feel your pain
Can you give me an example where you wouldn't have a maximum and a minimum?
what about a graph that only has the absolute maximum?
Wait, does absolute max and min necessarily mean a real max/min? I'm not sure if I made sense..
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
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.
if it's equal, f''(x) = 0 then it's neither a max or a min. It's just a constant
point of inflection is that particular area where the line changes from negative to positive. or vice versa
Oh! I mistook absolute min for a real min. Can yall help me with one more of this type?
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.
And a point of inflection is where the concavity of the line changes.
Thank you. Ya, I read up on the point of inflections, but never really grasped it. It's weird..
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.
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.
So A is wrong, because it is the opposite of what it should be
If there is a hip, there is a point?
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.
I'm sorry, a saddle point? @Tommynaut
By hip, I meant a cubed function
You might know a saddle point as a horizontal point of inflection (so f'(x) = 0 and f''(x) = 0).
I'm off now, hope I helped
You did, thank you
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|
\[\det(A- \lambda I) = 0 \] is what I used
Honestly, I have never seen those symbols before in my life
I think that saddle graph is a variation.. my book had what I drew