Calculus of a Single Variable, Chapter 2, 2.2, Section 2.2, Problem 68

The tangent line will touch one point of the original function. This means that:
kx^4 = 4x-1
We have two variables that we don't know.
Find the derivative of f(x) . The k is a constant and we can use the power rule.
f'(x) = 4kx^3
The slope of the tangent line is four, so we will plug that into this equation.
4 = 4kx^3
Divide four k on both sides.
1/x^3=k
Substitute k back into the first equation, .
(1/x^3)x^4 = 4x-1
x = 4x-1
-3x=-1
x= 1/3
Plug this back to to determine k.
k(1/3)^4 = 4(1/3)-1
k(1/81)= 4/3-1
k(1/81)=1/3
Multiply 81 on both sides.
k= 81/3 = 27