query: $f:\mathbb{R} \to \mathbb{R}$ we have $f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$ such function is polynomial of degree less than or equal to two.
positive_passages.docid: 24553
positive_passages.text: $f$ has the form $f(x) =ax^2+bx+c$.
negative_passages.docid: 18905
negative_passages.text: $f(a)=f(b)=f_{+}'(a)=f_{-}'(b)=0$ and $|f''(x)|\le M$. Show that $|f(x)|\le \frac{M(b-a)^2}{16}$.
negative_passages.docid: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 54403
negative_passages.text: Given $f : \Bbb{R} \to \Bbb{R}$ is a continuous function. To prove that $\int_{0}^{1}f(x)x^2 dx = \frac{f(c)}{3}$ for some $c \in [0,1]$
negative_passages.docid: 49121
negative_passages.text: Factorizing a polynomial $f$ in $A[x]$ (with $A$ commutative), where $f$ has a zero in its field of fractions
negative_passages.docid: 149328
negative_passages.text: A function such that $f:\mathbb C \to \mathbb C$ such that $f(z)f(iz)=z^2$ satisfies $f(z)+f(-z)=0$.(MADHAVA-2020)
negative_passages.docid: 121304
negative_passages.text: Is the set of polynomials of degree less than or equal to $n$ closed?
negative_passages.docid: 147184
negative_passages.text: $f:[-1,1]\to \mathbb{R}$ is a continuous function such that $f(2x^2-1)=(x^3+x)f(x).$ Find $\lim_{x\to 0}\frac {f(\cos x)}{\sin x}$
negative_passages.docid: 139111
negative_passages.text: Prove that $a<c<b$, $\frac{f(b)-f(a)-(b-a)f'(a)}{g(b)-g(a)-(b-a)g'(a)}=\frac{f"(c)}{g"(c)}$
negative_passages.docid: 24107
negative_passages.text: Bijection $f:\mathcal{P}(A)\to(A\to \{0,1\})$
negative_passages.docid: 65946
negative_passages.text: A function $f: \mathbb R \to \mathbb R$ satisfies equation $f(x+y)=f(x)f(y)$ for all $x,y\in \mathbb R $
negative_passages.docid: 19916
negative_passages.text: $f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$
negative_passages.docid: 65901
negative_passages.text: Solution verification to complex analysis problem, prove $f$ is a polynomial.
negative_passages.docid: 75266
negative_passages.text: $f(a+b)=f(a)+f(b)$ but $f$ is not linear
negative_passages.docid: 134043
negative_passages.text: Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $
negative_passages.docid: 2677
negative_passages.text: Baire: Show that $f\colon \mathbb{R}\to\mathbb{R}$ is a polynomial in an open bounded set
negative_passages.docid: 62110
negative_passages.text: Proving a function $\frac{1}{2y}\int_{x-y}^{x+y} f(t) dt=f(x)$ is a linear polynomial