Let $f(x)=x^6$ and $g(x)=2x^5-2x-1$. Find x such that these to functions are tangent.
My attempt:
$f(x)=g(x)\implies x^6-2x^5+2x+1=0$ and I know it's a polynomial and I could approximate it's roots but I have to find the exact $x$ such that they are tangent.
Or I thought their tangents to the graphic in that point must be equal so
$$f'(x)=g'(x)\implies 3x^5-5x^4+1=0.$$ and yet again another polynomial which does not have any nice roots.
Are there any other ways or am I just supposed to guess the solutions? Or approximate the solutions and the answers?
Let $P(x)=f(x)-g(x)=x^6-2x^5+2x+1$. Then, by the Euclidean algorithm,$$\gcd\bigl(P(x),P'(x)\bigr)=18(x^2-x-1).$$Therefore, the points that you are interested in are the roots of $x^2-x-1$.