Graphing All the Way: What Do the Signs of f(x), f′(x), and f′′(x) Tell Us About the Graph of f(x)?

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Info about the sign of $f(x)$	Impact on the graph of $f(x)$

$f(x) > 0$	$f(x)$ is above the $x$ -axis

$f(x) < 0$	$f(x)$ is below the $x$ -axis

$f(x) = 0$	$f(x)$ is on the $x$ -axis ( $x$ -intercept)


Info about the sign of $f'(x)$	Impact on the graph of $f(x)$

$f'(x) > 0$	$f(x)$ is increasing

$f'(x) < 0$	$f(x)$ is decreasing

$f'(x) = 0$	$f(x)$ is often turning around (max or min)

(a)

Assuming that $f'(x) > 0$ (i.e. $f(x)$ is increasing):


Info about the sign of $f''(x)$	Means this about $f'(x)$	Impact on the graph of $f(x)$

$f''(x) > 0$	$f'(x)$ (slopes of $f(x)$ ) are increasing (e.g., $3$ to $5$ to $8$ )	$f(x)$ is getting steeper

$f''(x) < 0$	$f'(x)$ (slopes of $f(x)$ ) are decreasing (e.g., $8$ to $5$ to $3$	$f(x)$ is getting shallower

$f''(x) = 0$		$f(x)$ often has an inflection point (shallower to steeper or vice versa)

(b)

Assuming that $f'(x) < 0$ (i.e., $f(x)$ is decreasing):


Info about the sign of $f''(x)$	Means this about $f'(x)$	Impact on the graph of $f(x)$

2024-10-10 13:50:51