Question

Find the number c if the line y=x is to be tangent to the graph of f(x)=x^4+c. what are the coordinates of the points of tangency?

Answer 1

so x = x^2 + c
x^2 - x + c = 0 must have only one root, that is,
the discriminant must be zero

b^2 - 4ac = 0
1 - 4(1)(c) = 0
c = 1/4

so x^2 -x + 1/4 = 0
(x- 1/2)^2 = 0
x = 1/2
then y = 1/2

the point is (1/2,1/2)