Q. 58 - 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 $f$.

$f\left(x\right)=x^2-x+100$.

Step-by-Step Solution

Answer

The graph of the function $f\left(x\right)=x^2-x+100$ is,

Step 1 . Given information

$f\left(x\right)=x^2-x+100$.

Step 2 . Consider the equation,

$f\left(x\right)=x^2-x+100$.

$y=x^2-x+100$

Now point table for the function is given by,

$x$	$y$	$\left(x,y\right)$
$0$	$100$	$\left(0,100\right)$
$-1$	$102$	$\left(-1,102\right)$
$1$	$100$	$\left(1,100\right)$
$10$	190	$\left(10,190\right)$
$-10$	$190$	$\left(-10,190\right)$

Step 3 . The graph of the function is,

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

$$\begin{gathered} \frac{d}{dx}\left(x^2-x+100\right)=0 \\ 2x-1=0 \\ 2x=1 \\ x=\frac{1}{2} \end{gathered}$$

Therefore, $f$ has a local minimum at $x=\frac{1}{2}$.

Step 5 . Therefore, the sign chart of f is shown below:

Therefore, the function $f$ is increasing on $\left(\frac{1}{2},\infty \right)$ and decreasing on $\left(-\infty ,\frac{1}{2}\right)$.

Again,

$$\begin{gathered} \underset{x\to -\infty }{\lim }f\left(x\right)=\underset{x\to -\infty }{\lim }\left(x^2-x+100\right) \\ =-\infty \\ \underset{x\to \infty }{\lim }f\left(x\right)=\underset{x\to \infty }{\lim }\left(x^2-x+100\right) \\ =\infty \end{gathered}$$

Therefore, the function is defined everywhere, it has no real roots. Positive on $\left(\frac{1}{2},\infty \right)$ and negative elsewhere. Local minimum at $x=\frac{1}{2}$.

The function $f$ is increasing on $\left(\frac{1}{2},\infty \right)$ and decreasing on $\left(-\infty ,\frac{1}{2}\right)$.

The limits are $\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=\infty$.