Mathematics

Problem

For the real parameter

k

, define the function

f_k:\mathbb R \to \mathbb R

f_k(x)=x^{3}+3x^{2}+k

. a. (i) Prove that

f_k

is continuous on

\mathbb R

. b. (ii) Evaluate

\displaystyle \lim_{x\to-\infty}f_k(x)

and

\displaystyle \lim_{x\to\infty}f_k(x)

. c. (iii) Using the results of (i) and (ii), show that

f_k

is surjective for every real value of

k