query: $|f(x)|\leq M/16(b-a)^2$?
positive_passages.docid: 18905
positive_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: 34888
negative_passages.text: An MCQ for finding the extremal of the functional $J = \int_{a}^{b} F(x, y, y^{'})$
negative_passages.docid: 56958
negative_passages.text: bounding a $C^1$ function that upholds $f'(t)\leq 1-f(t)^2$
negative_passages.docid: 148470
negative_passages.text: bound of the function
negative_passages.docid: 107727
negative_passages.text: $m\{x\in [0,1]:f'(x)=0\}>0$
negative_passages.docid: 25419
negative_passages.text: If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$
negative_passages.docid: 158180
negative_passages.text: An inequality of $\int_0^1 |f(x)|dx$
negative_passages.docid: 72856
negative_passages.text: Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
negative_passages.docid: 63526
negative_passages.text: Proving that $(\sup_{x\in R}|f'(x)|)^2\leq 4\sup_{x\in R}|f(x)|\cdot\sup_{x\in R}|f''(x)|$.
negative_passages.docid: 137614
negative_passages.text: $\left|\frac{x}{|x|}-\frac{y}{|y|}\right|\leq |x-y|$, for $|x|, |y|\geq 1$?
negative_passages.docid: 105641
negative_passages.text: Prove that $ab\leq\frac{1}{2\varepsilon}a^2+\frac{\varepsilon}{2}b^2$
negative_passages.docid: 91617
negative_passages.text: Prove this inequality $a \sqrt{1-b^2}+b\sqrt{1-a^2}\le1 $
negative_passages.docid: 69804
negative_passages.text: Prove that if $f$ is entire and $\vert f(z^2) \vert \leq 2 \vert f(z) \vert$, then $f$ is constant
negative_passages.docid: 6613
negative_passages.text: AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$
negative_passages.docid: 44653
negative_passages.text: $(\int f_1d\mu)^2+\cdots+(\int f_nd\mu)^2\leq(\int \sqrt{f_1^2+\cdots+f_n^2}d\mu)^2$
negative_passages.docid: 78599
negative_passages.text: How to prove that $\frac{b-a}{\sqrt{1-a^2}}<\sin^{-1}b-\sin^{-1}a<\frac{b-a}{\sqrt{1-b^2}}$ by using Mean Value Theorem?
negative_passages.docid: 138544
negative_passages.text: For which values of $x$, will $f (x) . f '(x) < 0$?