f(x)=ax^2+bx+c
f'(x)=2ax+b
f(0)=2 → c=2
f'(0)=1 → b=1
Also
f(x)=ax^2+x+2
f(w)=0 → aw^2+w+2=0 → a=(-w-2)/w^2
--> f(x)=(-w-2)/w^2 *x^2+x+2
f'(x)=2*(-w-2)/w^2 *x +1
2*(-w-2)/w^2 *xH +1=0
xH=w^2/(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:
\( \max \left\{\frac{(-w-2) x^{2}}{w^{2}}+x+2 \mid w=12\right\}=\frac{32}{7} \) at \( (w, x)=\left(12, \frac{36}{7}\right) \)
\( \max \left\{\frac{(-w-2) x^{2}}{w^{2}}+x+2 \mid w=15\right\} \approx 5.30882 \) at \( (w, x) \approx(15,6.61765) \)
\( \max \left\{\frac{(-w-2) x^{2}}{w^{2}}+x+2 \mid w=18\right\}=6.05 \) at \( (w, x)=(18,8.1) \)
:-)