examday

/TMUA

A differentiable function $f$ satisfies

for every real $x$, and $f(0)=0$.

Which statements must be true?

• I. The equation $f(x)=k$ has three distinct real solutions if and only if

  $\displaystyle |k|<\frac{16}{3}.$

• II. Whenever the three solutions are $\alpha,\beta,\gamma$,

  $\displaystyle \alpha^2+\beta^2+\gamma^2=24.$

• III. If

  $\displaystyle 0<k<\frac{16}{3},$

  then the middle of the three solutions is positive.