Consider the system of linear equations in x,y,z: [x+2y+tz=0, 6x+y+5tz=0, 3x+t^2y+f(t)z=0,…

Question

Consider the system of linear equations in $x,y,z$:
$$x+2y+tz=0,\qquad 6x+y+5tz=0,\qquad 3x+t^2y+f(t)z=0,$$
where $f:\mathbb{R}\to\mathbb{R}$ is a differentiable function. If this system has infinitely many solutions for all $t\in\mathbb{R}$, then:

Accepted Answer

For a homogeneous system to have infinitely many (non-trivial) solutions, the coefficient determinant must vanish for all $t$:
$$\det\begin{pmatrix}1&2&t\6&1&5t\3&t^2&f(t)\end{pmatrix}=0.$$

Expanding: $1\big(f(t)-5t^3\big)-2\big(6f(t)-15t\big)+t\big(6t^2-3\big)=0$, which simplifies to $-11f(t)+t^3+27t=0$, i.e.
$$f(t)=\frac{t^3+27t}{11}.$$

Then $f'(t)=\dfrac{3t^2+27}{11}=\dfrac{3(t^2+9)}{11}>0$ for all $t\in\mathbb{R}$.

So $f$ is strictly increasing on $\mathbb{R}$ --- Option B.

Solutions · 1