Chapter 4, Problem 2RE

To determine

To find: To find the global extreme values.

Answer to Problem 2RE

There are no global extreme values for this function.

Explanation of Solution

Given information:

The given function is ${\text{y = x}}^{\text{3}}{\text{ - 9x}}^{\text{2}}\text{ - 21x - 11}$

The interval for the given function is $\text{-}\infty \text{ < x < }\infty$

Calculation:

The given function is,

  ${\text{y = x}}^{\text{3}}{\text{ - 9x}}^{\text{2}}\text{ - 21x - 11}$

There may not be any global extreme values for the function since the domain is all real numbers rather than a closed interval.

Consider,

  $\underset{\text{x}\to \infty }{\text{lim}}\left({\text{x}}^{\text{3}}{\text{ - 9x}}^{\text{2}}\text{ - 21x}\right)=\infty$

There can be no global maximum because the function increases without bound.

  $\underset{\text{x}\to -\infty }{\text{lim}}\left({\text{x}}^{\text{3}}{\text{ - 9x}}^{\text{2}}\text{ - 21x}\right)=\text{ -}\infty$

There can be no global minimum because the function decreases without bound.

Therefore, the function has no global extreme values.