Question

The curve f(x)=-x^4+2x^2+x has a tangent at the point (1,2). Is this line tangent to the curve at another point and, if so where?

Answer 1

dy/dx = 4x^3 + 4x + 1
so at (1,2) dy/dx = 4+4+1 = 9

so is 4x^3 + 4x + 1 = 9 ?
x^3 + x - 2 = 0
we already know that (x-1) is a factor, so by synthetic division I found
x^3 + x - 2 = 0
(x-1)(x^2 + x + 2) = 0
x = 1, we knew that or
x = (-1±√-7)/2 which is not real

so there are no other points where the tangent has a slope of 9

Answer 2

find the tangent line, graph both of them