Question on second derivatives
its attached as a file Question on second derivatives
its attached as a file @Mathematics

Question

anonymous · Answer

attachment

turingtest · Answer

Can you find the inflection points?
How far can you get with this problem?
Please try.

anonymous · Answer

sure....hold on

anonymous · Answer

the second derivative is 8-16sin(2x)  and f''(0) is 8 and f''(pi) is also 8

turingtest · Answer

you are right about the derivative, but knowing f''(0) and f''(8) will not help us find concavity. do you know how to find inflection points?

turingtest · Answer

f''(pi)*

anonymous · Answer

isnt that where the concavity changes?

turingtest · Answer

right, and how do we find those points? (hint: the same way we find where a graph is increasing or decreasing using the first derivative, only here we do it with the second derivative)

turingtest · Answer

give up?

anonymous · Answer

huuuuhh yes

turingtest · Answer

we set f''(x)=0 and solve for x
those will be our inflection points
can you do that now?

anonymous · Answer

Inflection points are found when the Finding the F''(x)=0, however i'm not exactly sure what F'' is

turingtest · Answer

I'll prepare the answer while you work to have it ready in case you get lost.
@outkast   f''(x)=8-16sin(2x)

anonymous · Answer

No i mean't how F' measures change of slope, F'' measures the rate of changing the slope?

turingtest · Answer

F'(x) measures the slope of the tangent at x
F''(x) measures the rate of change of the slope at x (its "concavity")

anonymous · Answer

does that mean that F''' is the change of the rate of change of the slope XD

turingtest · Answer

Yes, I guess so. In physics it's called "Jerk", but in math it's just called the third derivative.

turingtest · Answer

$$f''(x)=8-16\sin(2x)$$$$\sin(2x)={1\over2}$$$$2x=\sin^{-1}(1/2)=\left\{ {\pi \over6}, {5\pi \over 6}\right\}$$$$x=\left\{ {\pi \over12},{5\pi \over12} \right\}$$so now we have three intervals to check: $$(0,{\pi \over12}) $$$$({\pi \over12},{5\pi \over 12}) $$and $$({5\pi \over 12}, \pi)$$We now need to pick a number N in each interval. 
If f''(N)<0 in that interval the graph is concave down. 
If f''(N)>0 in that interval the graph is concave up. 
Let's pick some a number N from the first interval. Why not try N=pi/24 ?

What is the value of f''(pi/24) ?

turingtest · Answer

the questioner appears to have left. how sad...

anonymous · Answer

How did you get 2x=sin^-1(1/2)?