Calculus and Algebra?

Question

I have two quick questions. The first one, I am stuck on and I keep getting zero, but I don't think its correct.

1. Find the intervals where f is concave upward and concave downward:
f(x)= x^3 -3x +1

As I previously mentioned, I keep getting zero when finding the intervals where the concavity is, but I don't think its correct. If someone could explain how to find it, I would greatly appreciated. I don't want the answer just an an explanation!

Secondly, this one is going to look stupid, but multiplying 'fractions' is my weakness. So if someone could confirm if I multiplied this correctly, I would also appreciated it!

2. (23- (x/500x) ) * (x)
my answer: (23x - (x^2 / 500x) )
huge possibility that its wrong (and probably is)! Not my strong suit, as previously mentioned.

So if someone could help, I would greatly appreciate it! Thanks and have a great day! :)

Answer 1

always remember that concavity means 2nd derivative

you have f(x)=x^3-3x+1
the derivative would be f'(x)= 3x^2-3
the 2nd derivative would be f"(x)= 6x

now you take the your double derivative and find your zero and that would be called your inflection point and that is where you can find out if your graph is concave up or down

so you have f"(x) = 6x
6x=0 (solve for x)
you get x=0....that is your possible inflection point

now you have to check by testing it
you have to see what the graph is doing before 0 and after 0 so you test two points.
u pick a number less then 0 and greater then 0 and plug into ur 2nd derivative

first, you would take -1 and plug it into f"(x)=6x
f"(x)=6(-1) which is -6
you get a negative number which it means it is facing down


then take 1 and plug it into f"(x)=6x
f"(x)=6(1) which is 6
you get a positive number which means it is facing up


in conclusion, you are going from down to up
which means it is concave up

hope this helps!

Answer 2

1. The concavity reverses at points where the second derivative is zero, and the only such point is at x = 0. Sketch a graph of the thing to get a better feel for this.
2. Simplify the inner parentheses: (x/500x) = 1/500. So the final result will be 22.998 x.

Answer 3

1)

2) x/500x = 1/500, so...

(23 - 1/500)x = 22 499/500 x = 22 998/1000 x = 22.998x

Answer 4

Note that the two answers you have listed in 2. are the same. One just has X factored out and multiplied by the rest of the answer.

Answer 5

i can't answer your question to small to answer your sum