Prove that $|a|+|b|+|c|\le17$ if $p(x)=ax^2+bx+c$ is a real polynomial with $|p|\le1$ for $0\le x\le1$

Let $f(x) = ax^2 + bx + c$ We know that $$ \left|\frac{a}2\right| = |f(0) + f(1) - 2f(0.5)| = |[f(0) - f(0.5)] - [f(0.5) - f(1)]|\\ \leq |f(0) - f(0.5)| + |f(0.5) - f(1)| \leq 2+2 = 4 $$ so $|a| \leq 8$. Clearly, $|c| = |f(0)| \leq 1$. That leaves $b$. We get $$ |b| = |4f(0.5) - f(1) - 3f(0)| \leq 3|f(0.5)-f(0)| + |f(0.5) - f(1)| \leq 3\cdot 2 + 2 = 8 $$ which is what we need.

It's also worth noting that $$ f(x) = 8x^2 - 8x + 1 $$ demonstrates that $17$ is a strict bound, so we cannot do any better.


Hint: Let $P$ your polynomial. Put $P(0)=u$, $P(1/2)=v$ and $P(1)=w$. Then $|u|, |v|, |w|$ are $\leq 1$. Find $a,b,c$ in function of $u,v,w$, and bound them.

Tags:

Quadratics

Related

Topology of a circle Do different methods of calculating fractional derivatives have to be equal? Prove $\left(\frac{a+1}{a+b}\right)^a+\left(\frac{b+1}{b+c}\right)^b+\left(\frac{c+1}{c+a}\right)^c \geqslant 3$ Why is the determinant defined in terms of permutations? A subtle limit question: $\lim _{n \to \infty}\sin(\pi\sqrt {n^2+0.5n+1})$ Prove that if fewer than $n$ students in class are initially infected, the whole class will never be completely infected. Prove that if V is finite dimensional then V is even dimensional? Sum of three consecutive prime numbers is $173$ Prove that $\int_{0}^{\infty}{1\over x^4+x^2+1}dx=\int_{0}^{\infty}{1\over x^8+x^4+1}dx$ The integral of a function on manifold and differential form Why is the probability of $\emptyset$ equal to 0? Is Floer homology always isomorphic to the singular homology of some space?

Recent Posts