Let f(x)=x3+3x+1 for all real x.
a. Verify that f is continuous and differentiable on R.
b. Compute f′(x) and determine its sign for every real x.
c. For any real numbers a<b, prove that f(a)<f(b). Conclude that f is one–to–one on R. Clearly state any theorems you invoke.
d. Show that the equation x3+3x+1=7 has exactly one real solution.