Question

the tangent line to the curve y=x^3-6x^2-34x-9 has slope 2 at two points on the curve. find the two points

Answer 1

$\bf y=x^3-6x^2-34x-9\implies \cfrac{dy}{dx}=3x^2-12x-34 \\\\\\ \stackrel{\textit{slope of 2}}{2}=3x^2-12x-34\implies 0=3x^2-12x-36 \\\\\\ 0=x^2-4x-12\implies 0=(x-6)(x+2)\implies x= \begin{cases} 6\\ -2 \end{cases}\\\\ -------------------------------\\\\ \stackrel{y}{f(6)}=(6)^3-6(6)^2-34(6)-9\implies f(6)=-213\\\\\\ (\stackrel{x}{6}~,~\stackrel{y}{-213}) \\\\\\ \stackrel{y}{f(-2)}=(-2)^3-6(-2)^2-34(-2)-9\implies f(-2)=27\\\\\\ (\stackrel{x}{-2}~,~\stackrel{y}{27})$