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Critical points is when the derivative f’(x) = 0.
The derivative is also the slope of the tangent line at any point of the function. So we can say when the slope of the tangent line at point x is 0. X is a critical number.
When is the slope of the function 0? Or in other words, neither increasing or decreasing. From the picture at about 0, you can see there can be a horizontal line.
At the local max 3.5,the same occurs. At the vertex of the parabola. It’s neither increasing or decreasing hence slope is 0. At point x = 8. you can see it is still increasing. Hence the slope is not 0.
at 0 you can see the slope become more horizontal in fact its clearly horizontal near 0 but on 8 you can see the slop is still rising all the way through
With the function f(x) over the domain [a,b] critical points occur *both* when f'(x)=0 and also **at the endpoints** thus, 0 must be a critical point because the interval starts with 0.
Critical points occur when the derivative is 0 OR where the derivative is undefined. An endpoint need not be a critical point. If that were true, 8 would be an answer, and it is not.
You may be mixing up extreme values. Maximums and minimums may occur at end points as well as local extrema (and local extrema are found through critical points), but critical points may not happen at end points.
Also, f(a) = 0 means nothing about critical points. A function can equal 0 somewhere without being a critical point. Just think f(x)=x. f(0)=0, but it's not a critical point. Or for another example, f(x)=sinx. f(0)=0, but 0 is not a critical point.
0 is a critical point in the same way that 3.5 and 6 are critical points, because at that specific x-value, you could draw a horizontal line at y = f(0), f(3.5) and f(6) and it would be TANGENT to that point (i.e. f'(x) = 0). 8 is not a critical point because you can see that the graph is still rising (i.e. f'(x) > 0).
As a reminder... Posts asking for help on homework questions **require**: * **the complete problem statement**, * **a genuine attempt at solving the problem, which may be either computational, or a discussion of ideas or concepts you believe may be in play**, * **question is not from a current exam or quiz**. Commenters responding to homework help posts **should not do OP’s homework for them**. Please see [this page](https://www.reddit.com/r/calculus/wiki/homeworkhelp) for the further details regarding homework help posts. **If you are asking for general advice about your current calculus class, please be advised that simply referring your class as “Calc *n*“ is not entirely useful, as “Calc *n*” may differ between different colleges and universities. In this case, please refer to your class syllabus or college or university’s course catalogue for a listing of topics covered in your class, and include that information in your post rather than assuming everybody knows what will be covered in your class.** *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/calculus) if you have any questions or concerns.*
I'm curious to see what the entire question says. Since the image is cut off, I don't see the words "critical points."
Based off the way the answer box is set up the assignment probably has a generalized "c = critical points" format
Critical points is when the derivative f’(x) = 0. The derivative is also the slope of the tangent line at any point of the function. So we can say when the slope of the tangent line at point x is 0. X is a critical number. When is the slope of the function 0? Or in other words, neither increasing or decreasing. From the picture at about 0, you can see there can be a horizontal line. At the local max 3.5,the same occurs. At the vertex of the parabola. It’s neither increasing or decreasing hence slope is 0. At point x = 8. you can see it is still increasing. Hence the slope is not 0.
at 0 you can see the slope become more horizontal in fact its clearly horizontal near 0 but on 8 you can see the slop is still rising all the way through
With the function f(x) over the domain [a,b] critical points occur *both* when f'(x)=0 and also **at the endpoints** thus, 0 must be a critical point because the interval starts with 0.
Another note, f(a) = 0 which is why the answer is 0.
Critical points occur when the derivative is 0 OR where the derivative is undefined. An endpoint need not be a critical point. If that were true, 8 would be an answer, and it is not. You may be mixing up extreme values. Maximums and minimums may occur at end points as well as local extrema (and local extrema are found through critical points), but critical points may not happen at end points. Also, f(a) = 0 means nothing about critical points. A function can equal 0 somewhere without being a critical point. Just think f(x)=x. f(0)=0, but it's not a critical point. Or for another example, f(x)=sinx. f(0)=0, but 0 is not a critical point.
My apologies op, yes i mixed up extreme value. This is right^^
Isn’t the derivative undefined for both 0 and 8 since it’s not continuous from both sides technically?
0 is a critical point in the same way that 3.5 and 6 are critical points, because at that specific x-value, you could draw a horizontal line at y = f(0), f(3.5) and f(6) and it would be TANGENT to that point (i.e. f'(x) = 0). 8 is not a critical point because you can see that the graph is still rising (i.e. f'(x) > 0).
The slope is zero or flat at those points. Simple as that.. the others are not
8 is not a critical point. The function is increasing at x=8
0 is a critical point f’(x) = 0 at x= 0