query: If $f(b)-f(a)=(b-a)f'(\frac{a+b}{2})$ prove that any such function is a polynomial of degree $2$
positive_passages.docid: 24810
positive_passages.text: Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
negative_passages.docid: 61782
negative_passages.text: How to prove that $f(x) = \frac{-2x+1}{(2x-1)^2-1}$ is one-to-one on $(0,1)$?
negative_passages.docid: 69008
negative_passages.text: $f'(x) = [f(x)]^{2}$. Prove $f(x) = 0 $
negative_passages.docid: 30204
negative_passages.text: Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
negative_passages.docid: 127056
negative_passages.text: How to prove that a polynomial of degree $n$ has at most $n$ roots?
negative_passages.docid: 41037
negative_passages.text: Prove that $f'(a)=\lim_{n\to\infty}\frac{f(b_n)-f(a_n)}{b_n-a_n}$ under some conditions
negative_passages.docid: 60348
negative_passages.text: How do I prove $F(a)=F(a^2)?$
negative_passages.docid: 69219
negative_passages.text: Prove there is a point $z\in[a,b]$ at which $f(z)=\frac{f(x_1)+f(x_2)+\cdots+f(x_k)}{k}$
negative_passages.docid: 57368
negative_passages.text: Prove that a positive polynomial function can be written as the squares of two polynomial functions
negative_passages.docid: 85739
negative_passages.text: Show that $f(x) = 0$ for all $x \in [a,b]$ given $|f'(x)| \leq C|f(x)| $
negative_passages.docid: 107133
negative_passages.text: Let f(x) be a polynomial satisfying $\lim_{x \to\infty} \frac{x^4f(x)}{x^8+1}=3,$ $f(2)=5 $ and $f(3)=10$, $f(-1)=2$ and $f(-6)=37$
negative_passages.docid: 79359
negative_passages.text: Show that $f$ is a polynomial if it's the uniform limit of polynomais
negative_passages.docid: 148486
negative_passages.text: Suppose $f$ is entire and $|f(z)| \leq A + B |z|^{3/2}$. Show that $f$ is a linear polynomial.
negative_passages.docid: 78834
negative_passages.text: if f(x) if differentiable and continuous, prove $\frac{af(a)-bf(b)}{a-b} = f(c) + cf'(c) $
negative_passages.docid: 65619
negative_passages.text: Does $f^{(n)} = 0$ imply that complex $f$ is a polynomial?
negative_passages.docid: 141184
negative_passages.text: Proof that a degree 4 polynomial has at least two roots
negative_passages.docid: 96116
negative_passages.text: Show $P(z) $ is a polynomial of degree $n-1$ interpolating an Analytic function.