<s>[INST] Find a line which is tangent to the curve $y=x^4-4x^3$ at 2 points How can I solve this? Should I set the two points as $a^4-4a^3$ and $b^4-4b^3$?
 [/INST] Suppose a line $y=mx+c$ is tangent to the curve $y=x^4-4x^3$ at two points. Then the difference between these two equations must be a quartic polynomial with two double roots:
$$x^4-4x^3-mx-c=(x-a)^2(x-b)^2$$
$$=x^4-2(a+b)x^3+(a^2+4ab+b^2)x^2-2ab(a+b)x+(ab)^2$$
Comparing coefficients, we have
$$2(a+b)=4$$
$$a^2+4ab+b^2=(a+b)^2+2ab=0$$
$$m=2ab(a+b)$$
$$c=-(ab)^2$$
The first two relations yield values for $a+b$ and $ab$:
$$a+b=\frac42=2$$
$$ab=-\frac{(a+b)^2}2=-\frac{2^2}2=-2$$
The last two relations allow $m$ and $c$ to be evaluated:
$$m=2\cdot-2\cdot2=-8$$
$$c=-(-2)^2=-4$$
Hence the line tangent to $y=x^4-4x^3$ at two points is $y=-8x-4$.
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