Let $a,b \in \mathbb R$. If for $|x| \le 1$, $|ax^2 + bx + c| \le 1$, find the maximum possible value of $|2ax+b|$ for $x \in [-1,1]$.
Asked
Active
Viewed 50 times
0
-
I'm sure this is a duplicate. – wythagoras Aug 12 '16 at 09:31
-
Intuitively, we want to stretch/translate a parabola that fits inside a square with vertices $(\pm 1, \pm 1)$, such that the slope of its tangent line is as steep as possible. Through some kind of symmetry, argue that the vertex must be on the $y$-axis. This forces the parabola to pass through $(0, -1)$ and $(\pm 1, 1)$ so that $a = 2$, $b = 0$, and $c = -1$. So the max slope is $4$. – Adriano Aug 12 '16 at 09:39
-
http://math.stackexchange.com/questions/1749912/suppose-that-a-function-fx-ax2bxc-where-a-b-c-are-real-constants-sati – wythagoras Aug 12 '16 at 09:40
-
To show how @Adriano got their intuition, notice that $(ax^2+bx+c)'=2ax+b$. – Frenzy Li Aug 12 '16 at 09:41