## arcticf0x 3 years ago Show that every point on the curve y=b sin (x/a), where the curve meets the axes of x, is a point of inflextion. Do i have to plainly show that x=0 is a POI or its asking for something else?

1. amistre64

inflection tends to be a found by a second derivative; and tested for cavage

2. arcticf0x

yeah i know that, but the language isnt very clear up there.

3. amistre64

looks like: -b sin(x/a)/a^2 is the second derivative to me

4. arcticf0x

yeah

5. amistre64

since b=0 is trivial; it looks to be what x/a = any mutiple of pi

6. arcticf0x

sorry didnt get that.

7. amistre64

how to "show" it? i aint got a clue

8. amistre64

sin(n*pi) = 0 so when x/a = n*pi we are at zero

9. arcticf0x

yeah thats where i came, so is that the answer fianlly, x=0

10. amistre64

x = a n*pi would seem to be the answer to me; but since n is an arbitrary integer then a*n would have to be an interger as well.

11. amistre64

other than that; I got no idea what the "answer" might entail ....

12. arcticf0x

umm, it says, where the curve meets the axes of x, so y=0 at that point, this only gives x=0.

13. phi

inflection point is where the second derivative changes sign. we know that y=b sin (x/a) crosses the x-axis when x/a is some multiple of 2pi the 2nd derivative at these same points is -b sin(x/a)/a^2 = 0 also, if we go a dx distance below x/a and a little distance dx above x/a, the 2nd derivative changes sign. So we are at an inflection point.

14. arcticf0x

That seems clear, understood, thank you phi and amistre for your time :)

15. phi

*crosses the x-axis when x/a is some multiple of pi