Find the number of real solutions to the equation $ x^7=x+1$. The equation $x^7 = x+1 $ has:

a) no real solution
b)  no positive real solution
c) a real solution in the interval (0,2)
d) a real solution but not within (0,2)
Which is the correct answer and why ? How do i find the answer to such questions? 

Consider the 2 equations:
$$y = x^7 \tag{A}$$
$$y = x + 1 \tag{B}$$
What does equation (A) look like?  It has an intersection point at $(0, 0)$, grows incredibly fast before $x=-1$ and after $x=1$.
Equation (B) grows so much slower that it can only intersection (A) at one point.  Since $1^7 < 1 + 1 < 2^7$, the intersection point is somewhere between $x=1$ and $x=2$.  So the answer is $c$.
