Answer
The values of c are -$\sqrt \frac{8}{3}$ $\lt$ c $\lt$ $\sqrt \frac{8}{3}$
Work Step by Step
This question involves using the discriminants of quadratic expressions in order to determine the values. Since f(x) is a fraction with x in both the numerator and the denominator, the denominator cannot equal to zero for f(x) to be defined since dividing by zero is undefined.
Hence, the denominator $x^{2}+3cx+6$ must be always positive such that it does not touch the x-axis at all and hence does not equal to zero for all x values.
For this to be true, set the descriminants of the quadratic expression to be less than zero (Set $b^{2}-4ac \lt 0$ where a,b,c are the coefficients of the quadratic expression)
∴ (3c)$^{2}$-4(1)(6)$\lt$0
9c$^{2}$-24$\lt$0
c$^{2}\lt\frac{8}{3}$
∴-$\sqrt\frac{8}{3}\lt$c$\lt\sqrt\frac{8}{3}$
