Q. 63 - Calculus

Question

Sketch careful, labeled graphs of each function $f$ in Exercises $57-82$ by hand, without consulting a calculator or graphing utility. As part of your work, make sign charts for the signs, roots, and undefined points of $f$ and $f^{\text{'}}$and examine any relevant limits so that you can describe all key points and behaviors of f .

$f\left(x\right)=(2x + 11)(x^2+10)$.

Step-by-Step Solution

Answer

The graph for the function $f\left(x\right)=\left(2x+11\right)\left(x^2+10\right)$ is,

Step 1 . Given information

$f\left(x\right)=(2x + 11)(x^2+10)$.

Step 2 . Let f ( x ) = ( 2 x   +   11 ) ( x 2 + 10 ) .

Now point table for the function is given by,

$x$	$y$	$\left(x,y\right)$
$0$	$110$	$\left(0,110\right)$
$1$	$143$	$\left(1,143\right)$
$-1$	$99$	$\left(-1,99\right)$
$2$	$210$	$\left(2,210\right)$
$-2$	$98$	$(-2,98)$

Step 3 . The graph for the function is,

Step 4 . Now for critical point, f ' x = 0 .

$$\begin{gathered} \frac{d}{dx}\left(2x^3+11x^2+20x+110\right)=0 \\ 6x^2+22x+20=0 \\ 3x^2+11x+10=0 \\ 3x^2+5x+6x+10=0 \\ x(3 x+5)+2(3 x+5)=0 \\ (3 x+5)(x+2)=0 \\ x=-\frac{5}{3},-2 \end{gathered}$$

Therefore, $f$ has a critical point at $x=-\frac{5}{3},-2$. But that is a local maximum at $x=-2$ and local minimum $x=-\frac{5}{3}$.

Step 5 . The sign chart of f is shown below:

Therefore, the function $f$ is increasing on $\left(-\infty ,-2\right)$ and $\left(-2,-\frac{5}{3}\right)$ that $f$ is decreasing on elsewhere.

Again,

$$\begin{gathered} \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }\left(2x^3+11x^2+20x+110\right) \\ =-\infty \\ \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(2x^3+11x^2+20x+110\right) \\ =\infty \end{gathered}$$

Therefore, the function is defined everywhere, roots at $x=-\frac{11}{2}$. Positive on $\left(-\frac{11}{2},\infty \right)$ and negative elsewhere. Local minimum $x=-\frac{5}{3}$ and local maximum at $x=-2$.

The function $f$ is increasing on $(-\infty ,-2)\cup \left(-2,-\frac{5}{3}\right)$ and decreasing elsewhere. And the limits are $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty ; \underset{x\to \infty }{\lim }f\left(x\right)=\infty$.