f(x)=ax2+bx+c
f'(x)=2ax+b
f(0)=2 → c=2
f'(0)=1 → b=1
Also
f(x)=ax2+x+2
f(w)=0 → aw2+w+2=0 → a=(-w-2)/w2
--> f(x)=(-w-2)/w2 *x2+x+2
f'(x)=2*(-w-2)/w2 *x +1
2*(-w-2)/w2 *xH +1=0
xH=w2/(2w+4)
w=12 → xH=144/28=36/7
yH=-7/72 * (36/7)2+36/7+2
=50/7-18/7=32/7=4,57...
Wolframalpha zur Kontrolle:

max{w2(−w−2)x2+x+2∣w=12}=732 at (w,x)=(12,736)
max{w2(−w−2)x2+x+2∣w=15}≈5.30882 at (w,x)≈(15,6.61765)
max{w2(−w−2)x2+x+2∣w=18}=6.05 at (w,x)=(18,8.1)
:-)