$$ f(x)=ax^4+bx^3+cx^2+dx+e$$
$$ f'(x)=4ax^3+3bx^2+2cx+d$$
$$ f''(x)=12ax^2+6bx+2c$$
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$$ f(x)=0; f'(x)=0; f''(x)=0\Rightarrow c=d=e=0 $$
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$$ f(x)=ax^4+bx^3$$
$$ f'(x)=4ax^3+3bx^2$$
$$ f''(x)=12ax^2+6bx$$
$$f'(6)=0 \Rightarrow 4a\cdot 6^3+3b\cdot 6^2=0 \Rightarrow b=-8a$$
$$ f(x)=ax^4-8ax^3$$
$$ f'(x)=4ax^3-24ax^2$$
$$ f(x_1)=0=ax_1^4-8ax_1^3 =ax_1^3(x_1-8) \Rightarrow x_1=8$$
$$f'(x_1)=-8 =4a\cdot8^3-24a\cdot 8^2$$
$$-8 =2048a-1536a$$
$$-8=512a \Rightarrow a=-\frac{1}{64} \Rightarrow b=\frac{1}{8}$$
$$ f(x)= -\frac{1}{64}x^4+\frac{1}{8}x^3$$
