Thread: maximum problem

1. maximum problem

given f(x)=integration of sin(s^2)ds from 0 to x

i found out d(f(x))=sin(x^2)

i am unable to find the maximum for triple derivative of f(x) in interval [pi/2] please help

2. Re: maximum problem

so you want the maximum of $\dfrac {d^2}{{dx}^2}\left(\sin(x^2)\right)$ ?

$\dfrac d {dx}\left(\sin(x^2)\right) = 2x \cos(x^2)$

$\dfrac d {dx} \left(2x \cos(x^2)\right) = 2\cos(x^2) - 4x^2\sin(x^2)$

Taking a quick look at this on $0 \leq x \leq \dfrac {\pi} 2$

You can see that it looks like the maximum is at $x=0$

Checking for roots of $2\cos(x^2) - 4x^2\sin(x^2)=0$ we find

$x=0, x\approx 1.47466$

The second root is a minimum and so the maximum on this interval is at $x=0$

thanks