If $f'(x)=[f(x)]^2 \enspace \forall x \ in \mathbb{R}$ and $f(0)=0$ then $f=0$ everywhere.
69008
$f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
23615
Why does zero derivative imply a function is locally constant?
57026
$f(0)=f'(0)=f'(1)=0$ and $f(1)=1$ implies $\max|f''|\geq 4$
33989
If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
48592
Prove that $f'$ exists for all $x$ in $R$ if $f(x+y)=f(x)f(y)$ and $f'(0)$ exists
76285
Suppose $f$ is entire and $|f(z)| \leq 1/|Re z|^2$ for all $z$. Show that $f $ is identically $0$.
103706
If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
71953
If $\int_0^1 f x^n dx=0, \forall n \in \mathbb{N} $, then $f\equiv 0.$
131793
Zero integral of measurable $f$ on every interval implies $f=0$?
136980
Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.
74100
How do you solve $f'(x) = f(f(x))$?
58901
Why is every such function constant: $f(x)=f(x^2)$ for $x \in [0,1]$?
5180
How to show $\alpha\int_a^cf(x)dx+\beta \int_c^b f(x)dx =0$ means $f(x)=0$ everywhere $?$
144745
Proof of: If $x_0\in \mathbb R^n$ is a point of local minimum of $f$, then $\nabla f(x_0) = 0$.
41173
If $f\in S_\infty$ and $\int_{\mathbb{R}}x^pf(x)d\mu=0$ for all $p\in\mathbb{N}$ then $f\equiv 0$?
13202
Prove that there exists at least one $x_0\in\mathbb{R}$, such that $f(x_0)+f''(x_0)=0$
95260
Let $m$ be Lebesgue measure and $a \in R$. Suppose that $f : R \to R$ is integrable, and $\int_a^xf(y) \, dy = 0$ for all $x$. Then $f = 0$ a.e.