Recovering a curve from its curvature function

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Problem 3.1.16 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.1 "Differential Geometry of Curves"

Let α(s)\alpha (s) be a C2C^2 curve in R3\mathbb {R}^{3} parameterized by arclength. Suppose that for some function f(s)f(s), α′′(s)=f(s)α(s)\alpha ''(s) = f(s)\alpha (s). What can you deduce about f(s)f(s), α(s)\alpha (s)?

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Problem 3.1.21 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.1 "Differential Geometry of Curves"

Sketch the closed regular plane curve β:[−π,π]→R2\beta : [-\pi , \pi ] \to \mathbb {R}^{2} having β(0)=(0,0)\beta (0) = (0,0) and β′(0)=(1,0)\beta '(0) = (1,0), if β\beta's curvature function k(s)k(s) (s=s = arclength) is odd, satisfies

∫0πk(s) ds=3π/2, \int _0^\pi k(s) \, ds = 3\pi /2,

and has the following graph. (Include an explanation with your sketch.)

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References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.