Chapter 2, Problem 14RE

To determine

To find: The left-end and right-end behaviour of given polynomial function.

Answer to Problem 14RE

  $\begin{array}{l}x\to -\infty ;f(x)\to -\infty \\ x\to \infty ;f(x)\to \infty \end{array}$

Explanation of Solution

Given information:

A polynomial function is given as, $f(x)=\frac{1}{2}{x}^3+2x$

Calculation:

Given polynomial function is, $f(x)=\frac{1}{2}{x}^3+2x$ .

Comparing with standard form of a polynomial function $f(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{...}+a_{1}x+a_0$ .

Here, leading term of the polynomial is $\frac{1}{2}{x}^3$ .

So, $n=3=\text{odd}$ and $a_n=\frac{1}{2}>0$ .

Therefore, from the Leading Coefficient Test graph of the given function will fall to the left and rise to the right. i.e.

  $\begin{array}{l}x\to -\infty ;f(x)\to -\infty \\ x\to \infty ;f(x)\to \infty \end{array}$