Exercises: Derivatives 4 – "Math for Non-Geeks"

Hint

Since ${\displaystyle f''=(f')'=(\omega g)'=\omega g'=-\omega ^{2}f}$ and analogously ${\displaystyle g''=-\omega ^{2}g}$ both the functions ${\displaystyle f}$ and ${\displaystyle g}$ satisfy the differential equation ${\displaystyle f''+\omega f=0}$ (or ${\displaystyle g''+\omega g=0}$).

Hint

Similarly, we can show that ${\displaystyle f}$ is strictly monotonically decreasing in the cases ${\displaystyle a=b=0}$ and ${\displaystyle c<0}$, as well as for ${\displaystyle a<0}$ and ${\displaystyle 3ac>b^{2}}$.