9. (6 points) Consider the following information about the...

9. (6 points) Consider the following information about the graph of a continuous function $m(x)$: \begin{itemize} \item $\lim_{x \to -\infty} m(x) = -2$ and $\lim_{x \to \infty} m(x) = 2$ \item The table below summarizes when $m'$ and $m''$ are positive and negative, but with some information unknown: \begin{tabular}{|c|c|c|c|c|c|} \hline $x$ & $x < -2$ & $-2 < x < -1$ & $-1 < x < 0$ & $0 < x < 1$ & $1 < x < 2$ & $2 < x$ \\ \hline $m'(x)$ & $-$ & $-$ & $+$ & $+$ & $-$ & $-$ \\ $m''(x)$ & $?$ & $+$ & $+$ & $-$ & $-$ & $?$ \\ \hline \end{tabular} \end{itemize} Copy or print the axes below and sketch a graph of $m(x)$ with $-5 < x < 5$ that matches the information.

Transcript

00:01 We're told that we have this function q, and we're told that the limit, that the limit of q of x, as x goes to negative infinity, is equal to negative 2, and the limit of q of x, as x goes to positive infinity, is equal to 2.

00:29 And then we're given some intervals here.

00:32 So we're given some intervals.

00:39 And i'm just going to go ahead and do these on a number line.

00:42 So from negative 2, let me try that again.

00:55 Here we go.

00:58 Negative 2, negative 1, 0, 1, and 2.

01:09 All right, and then we're told we have q prime.

01:14 So for those values less than 2, q prime is negative.

01:19 For values between negative 1, negative 2 and negative 1, q prime is negative.

01:23 Between negative 1 and 0, we have positive, and the next one is positive.

01:27 And the next one is negative and the next one is negative and then we're told q double prime so we don't know what it is out here and then we're positive positive negative negative and we don't know out here and we're asked to graph this so i'm going to start with these two so the limit as x of sorry if q of x as x goes as x goes to negative infinity is going to be negative two so i'm just going to put a horizontal and last method out here and i'm going to stop there i could keep going but i know that i'm looking at negative infinity and i'll put one out here up at two as x goes to positive infinity so i know that the end behavior of the graphs are going to approach that and then so let's talk about this so q prime q prime is negative between from negative infinity up to negative two and it's negative from negative to negative one and so therefore that means means it's decreasing.

02:37 That means the function is decreasing, right? at 2, we don't know the concavity right here, but at 2, it turns a concave up.

02:49 So that means it has to be decreasing right up until negative 1.

02:54 So a negative 1 is going to start increasing.

02:58 And at negative 2, the concavity is going to change.

03:01 So i'm going to say it's going to look something like this.

03:08 It's going to be down.

03:13 And then at 2, it's going to, it's going to, still decrease it still decrease here but our concavity is going to change until we hit a minimum point right here so when we change from decreasing to increasing at negative one it's going to be an increasing there so and then we're going to increase from negative one all the way to one so we're going to increase from negative one all the way to one and then we need to decrease after that so my increase is going to have to take me above this let me switch back to green here.

03:53 Well, stay in the red.

03:55 So we see where...

03:56 So increase all the way up to one.

04:09 And i'm also at zero, i change concavity.

04:14 At zero, i changed concavity...

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