Achsensymmetrisch: \(f(x)=ax^4+bx^2+c\)
\(f'(x)=4ax^3+2bx\)
Nullstelle bei x=2: \(f(2)=0=16a+4b+c~~~~~(I)\)
Hochpunkt H(1|9):
\(f(1)=9=a+b+c~~~~~(II)\)
\(f'(1)=0=4a+2b\Rightarrow b=-2a\)
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$$ ~(I)~~~0=16a-8a+c\Rightarrow 0=8a+c ~~~~~(III)$$
$$ (II)~~~9=a-2a+c\Rightarrow 9=-a+c ~~~~~(IV)$$
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$$ (III)-(IV)~~~-9=9a\Rightarrow a=-1 $$
$$ (IV)\Rightarrow c=8$$
$$ b=2 $$
$$ f(x)=-x^4+2x^2+8$$