Consider the three linear equations with respect to $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ as
$$
\begi

Question

Consider the three linear equations with respect to $\mathrm{X}, \mathrm{Y}, \mathrm{Z}$ as
$$
\begin{aligned}
& (x) X+Y+2 Z=0 \
& (f(x)) X+3 Y+x^2 Z=0 \
& (5 x) X+6 Y+Z=0
\end{aligned}
$$having non-trivial solution for $X, Y, Z$. The curve $f(x)$ :

Professor PC Thomas Classes · Answer

We must have
$$
\begin{aligned}
& \left|\begin{array}{ccc}
x & 1 & 2 \
f(x) & 3 & x^2 \
5 x & 6 & 1
\end{array}ight|=0 \
& 	herefore x\left(3-6 x^2ight)-1\left(f(x)-5 x^3ight)+2(6 f(x)-15 x)=0 \
& 	herefore f(x)=\frac{x^3+27 x}{11} \quad \Rightarrow f^{\prime}(x)=\frac{3 x^2+27}{11}>0 \forall x \in R
\end{aligned}
$$