Check if functions are linearly independent

Question

I have 2 functions:

$$\tag{1} f_1(x) = \cos x – 2x$$
$$\tag{2} f_2(x) = x^2 \sin x$$

Where $f_1,f_2:\Bbb R \to \Bbb R$

And I need to check whether they are linearly dependent or not.

*NOTE: I didn't learn about Wronskian

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So I can take 2 scalars in R, $a_1,a_2 \in \Bbb R$ and check the solutions for:
$$a_1(\cos x – 2x) + a_2(x^2 \sin x) = 0$$
So I think I can take like: $a_1 = -\frac{x^2 \sin x}{\cos x -2x}, a_2 = 1, x = \pi \Rightarrow a_1 = 0$.

But it seems that i could even take: $a_1 = 0, a_2 = 1, x=\pi$ and it would hold.

So the scalars are non zero, the linear combination is zero, so the functions are linear dependent, did i made any mistake? (It seems to me a little bit weird)

Is it ok that I chose a specific $x$ for the solution or I somehow should show it $\forall x \in \Bbb R?$

if $\forall x$, I could still take: $a_1 = -\frac{x^2 \sin x}{\cos x -2x}, a_2 = 1$ and it would hold, no?

Answer 1

Hint: take different values of $x$ (for example $x=\pi, x={\pi\over 2}$) in the equation to show that $a_1=a_2=0$.

For $x=\pi, a_1(-1-2\pi)=0$ implies $a_1=0$, $x={\pi\over 2}$ implies that $a_2=0$