I have got this question
Suppose that $f$ is twice differentiable at every $x\in\mathbb{R}$ and that for every $x\in\mathbb{R}$ $$f''(x) + f(x) = 0.$$ Show that if $f(0) = 0, f'(0) = 0, $ and $\left|f(x)\right|\leq 24 $ for all $x\in\mathbb{R}$, then for every $x\in\mathbb{R}$ $$\left|f(x)\right|\leq x^{4}$$
As all the example in my notes does given the equation of $f(x)$ so I have got no idea to deal with question that with unknown $f(x)$. Would you guys give a brief introduction about how to deal with this kind of question and show me details step in order to help me to learn it in details?
p.s. I am currently learning Taylor Approximation and Error Estimation in the section of Differentiation and Its Applications
Hint:
$f$ is four times differentiable, $f^{(3)}(x) = -f^{(1)}(x)$ and $f^{(4)}(x) = f(x)$ for all $x$.
$f^{(n)}(0) = 0$ for $0\leq n\leq 3$.
$|f^{(4)}(x)| = |f(x)| \leq 24$ for all $x$.
For any given $x$ Taylor's formula (as extension of Mean Value Theorem) provides a point $\zeta_x$ (between $0$ and $x$) such that: $$ |f(x)| = \left|\frac{1}{24} f^{(4)}(\zeta_x)x^4\right| $$