Chapter 3: Q. 58 (page 261)

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 $f^{'}$ and examine any relevant limits so that you can describe all key points and behaviors of $f$ .
$f (x) = x^{2} - x + 100$ .

Short Answer

Expert verified

The graph of the function $f (x) = x^{2} - x + 100$ is,

Step by step solution

Step 1. Given information

$f (x) = x^{2} - x + 100$ .

Step 2. Consider the equation,

$f (x) = x^{2} - x + 100$ .

$y = x^{2} - x + 100$

Now point table for the function is given by,

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

Step 3. The graph of the function is,

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

$\frac{d}{d x} (x^{2} - x + 100) = 0 2 x - 1 = 0 2 x = 1 x = \frac{1}{2}$

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 $(\frac{1}{2}, \infty)$ and decreasing on $(- \infty, \frac{1}{2})$ .

Again,

$\lim_{x \to - \infty} f (x) = \lim_{x \to - \infty} (x^{2} - x + 100) = - \infty \lim_{x \to \infty} f (x) = \lim_{x \to \infty} (x^{2} - x + 100) = \infty$

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

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

The limits are $\lim_{x \to - \infty} f (x) = - \infty$ and $\lim_{x \to \infty} f (x) = \infty$ .

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Consider the equation,

Step 3. The graph of the function is,

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

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

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