Let for all real .

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

f(x)=x^{3}+3x+1

for all real

x

. (1) Verify that

f

is continuous and differentiable on

\mathbb R

. (2) Compute

f'(x)

and determine its sign for every real

x

. (3) For any real numbers

a<b

, prove that

f(a)<f(b)

. Conclude that

f

is one–to–one on

\mathbb R

. Clearly state any theorems you invoke. (4) Show that the equation

x^{3}+3x+1=7

has exactly one real solution.