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Sketch the graph of a function that satisfies all of the given conditions

$ f'(x) > 0 $ if $ x \not= 2 $, $ f"(x) > 0 $ if $ x < 2 $,$ f"(x) < 0 $ if $ x > 2 $, $ f $ has inflection point $ (2, 5) $,$ \displaystyle \lim_{x\to\infty} f(x) = 8 $, $ \displaystyle \lim_{x\to-\infty} f(x) = 0 $

$f^{\prime}(x)>0$ if $x \neq 2 \Rightarrow f$ is increasing on $(-\infty, 2)$ and $(2, \infty)$$f^{\prime \prime}(x)>0$ if $x<2 \Rightarrow f$ is concave upward on $(-\infty, 2)$$f^{\prime \prime}(x)<0$ if $x>2 \Rightarrow f$ is concave downward on $(2, \infty)$$f$ has inflection point (2,5)$\Rightarrow f$ changes concavity at the point (2,5)$\lim _{x \rightarrow \infty} f(x)=8 \Rightarrow f$ has a horizontal asymptote of $y=8$ as $x \rightarrow \infty$$\lim _{x \rightarrow-\infty} f(x)=0 \Rightarrow f$ has a horizontal asymptote of $y=0$ as $x \rightarrow-\infty$

01:56

Fahad P.

Calculus 1 / AB

Calculus 2 / BC

Chapter 4

Applications of Differentiation

Section 3

How Derivatives Affect the Shape of a Graph

Derivatives

Differentiation

Volume

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Boston College

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So we're told that for X being not equal to two are derivative is always positive. So we're increasing at all points of X that are not equal to two. And we're also told that we are concave up or a second derivative is greater than zero for X being less than two. And then our second derivative is less than zero for X being greater than two hour. Concave down after two. So we have this inflection point at two comma five. And then we also have told that we have the limit as X goes to infinity um being equal to eight in the limit as X goings to negative infinity of fx being equal to zero. So we have two horizontal assam totes at eight and zero. And that's what I've drawn here. Can say eight. Put this in black. Eight. Sleep. Okay. So we have these two horizontal assam totes at eight and zero. And we know that as we go to infinity we're going to be getting close to this top one and as we go to negative infinity we're gonna be getting close to this bottom one. So if we look at our um if you look at our conditions again we're increasing for all x not equal to two. And we are concave up for X being less than two. So we know we're increasing in concave up for X less than two. So I'm gonna put a point here too and I'm actually going to draw Or just put a point there too and this is at 2:05, so this is five and we know that before too we're concave up and increasing so we're gonna be looking something like this And then after two were still increasing but where concave down. So we're now gonna be concave down like that but we're still increasing and we're going towards our horizontal asan totes on both sides. So as we go farther and farther in the negative X. Direction, we're getting closer and closer to zero. And as we go further and further in the positive X. Direction, we're getting closer and closer to this ASEM towed at X is equal to or sorry, why is equal to eight.

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