Answer
$0.000636$
Work Step by Step
$f_x=-\sqrt 2 \sin x \sin (y+z) \\ \implies f_x(0,0,\dfrac{\pi}{4})=-\sqrt 2 \sin (0) \sin (0+(\pi/4))=0$
Next, $f_y=\sqrt 2 \cos x \cos (y+z) \\ \implies f_y(0,0,\dfrac{\pi}{4})=-\sqrt 2 \sin (0) \sin (0+(\pi/4))=0$ and $f_z=2y-3x \implies f_z(1,1,0)=2y-3x=2(1)-3(1)=-1$
The error can be found as:
$|E(x,y,z)| \leq \dfrac{1}{2} \times (\sqrt 2) [ |x-0| +|y-0|+|z-\dfrac{\pi}{4}|)^2$
$\implies E \leq \dfrac{\sqrt 2}{2} \times (0.01+0.01+0.01)^2 =0.000636$
