Question

b) Find f''(x). Find the intervals of concavity. Locate inflection points.

Answer 1

\[f''(x) = \frac{ 50-30x^2 }{ (x^2+5)^3 }\]

Answer 2

The only intervals of concavity I figured out was \[(-\infty,0)(0,\infty) \] and both were concave upwards when I plugged it back into f''(x). However I have a feeling that I messed up somewhere and that either I'm missing a point or something.

Answer 3

So I need to make the numerator and denominator of f''(x) = 0 and solve for both?

Answer 4

will be concave where f'' is positive. Inflection points , where f''=0

Answer 5

since the denominator is always positive, it will depend on nominator.
50 -30x^2=0
x^2=5/3, or x=+-sqrt(5/3). This are inflection points.
for x inside this interval f'' is positive; outside - negative. So...

Answer 6

The concept is that f"(x) measures the rate of change of slope of a function. You need to figure out for what values of "x" the rate is negative or positive. And inflection is the point on the where f"(x)=0

Answer 7

Oh ok! I see now. For some reason I was doing 50-30(0)^2 instead of 50-30x^2 = 0. Looking down at my messy sheet of paper I had the sqr(5/3) but I simply dismissed it for some reason. So just to be sure, when making f''(x) = 0 to find inflection points should I always use the numerator when I'm doing it with rational functions?

Answer 8

I misread the question, you don't need to figure out the asymptotes for concavity.

Answer 9

depends on each case. In this case the denominator doesn't afect the sign, so it is irrelevant for the question

Answer 10

Ahh okk myko.

And that's fine imer, that's actually another part of the question I wasn't sure about so you indirectly helped me out!

And to figure out concavity I need to solve f''(sqr(5/3)) and f''(-sqr(5/3)) right?

Answer 11

well wait I meant I need to make a "sign table" and test plot points around those two points, right?

Answer 12

concavity is in the interval [-sqr(5/3), sqr(5/3)]. That's where f'' is positive

Answer 13

yes

Answer 14

Understand it now! Thanks you two!

Answer 15

yw