How do I obtain this graph? I keep getting stuck.

[pallid falcon]

When finding the intervals where the graph is either increasing or decreasing, I’m confused about intervals [-2π, -π), (-π, 0). The first yields a negative number and the second interval yields a positive number, but the graph shows otherwise. Why?

Also, how do I obtain the roots of the second derivative in order to find the concavities? I keep getting 0, but I’m not able to obtain any more roots. The second derivative is y”=2sin(x)+4xcos(x)-(x^2)sin(x).

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[full mesa]

The graph is right. Maybe ask your graphic tool to plot sin against your curve to see what happens.

[pallid falcon]

I know. But that’s the correct answer at the back of the book. My question is, how do I get that graph?

I explained the issues I was having in the body of my text. I’m not able to obtain all of the roots.

[heady ridge]

x^2sin(x)

well that graph looks correct lmao, but finding the critical points...

wait bro what

why the hell do you want the roots of the second derivative-

[crimson locust]

why does the first yield a positive and second a negative?

[heady ridge]

hmmm, you're wrong there
remember the graph of sin(x)

[pallid falcon]

I plugged in some values within those intervals into f’ and that’s what I got

[heady ridge]

um, they're asking you to plot x^2sin(x)

[pallid falcon]

I’m trying to use the first and second derivative tests to plot it

[crimson locust]

the values of f' will only tell you the gradients at those points though

[heady ridge]

you have them confused 😭

[pallid falcon]

Increasing or decreasing?

[crimson locust]

to find positive/negative, you'll need the actual graph

[pallid falcon]

Wdym?

[heady ridge]

you don't have to find the values of f'' where it is zero

you have to find where f' is 0, or the first derivative

then classify them as inflection, minima or maxima

[pallid falcon]

That’s what I did. f’ is zero at x=0, which forms part of an interval

[crimson locust]

if you look at the example graph, you will see that for a lot of the region [-π, 0], you can see that most of it does have increasing gradient

it's still under the x-axis

[heady ridge]

where else is f' 0?

there are more points

$2x\sin(x) + x^2\cos(x) = 0$

[maiden cloud]

Ravi #NoLifer

[heady ridge]

$2\sin(x) = -x\cos(x)$

[maiden cloud]

Ravi #NoLifer

[heady ridge]

$-2\tan(x) = x$

[maiden cloud]

Ravi #NoLifer

[heady ridge]

that looks easy enough with a calculator

[crimson locust]

i just think that there is an easier way to plot a graph

[heady ridge]

tbh what do you think it is?

we need the maxima and minima of the graph

the zeroes are simple

[pallid falcon]

I’ll show you more of the work I’m trying

[crimson locust]

i mean of course you'll need to use the derivative graph

[crimson locust]

but just drawing a sign diagram will give you a rough idea of where everything is

[heady ridge]

this should have 3 solutions in the interval we're working with

[crimson locust]

find the maxima/minima and then maybe test a few other key values

[heady ridge]

find the maxima/minima is the hard part lmao

[heady ridge]

this is transcedental

[crimson locust]

you're not allowed a calculator

hm

[heady ridge]

bro we have 5 solutions

bruhhh

[heady ridge]

shitty taylor series 🔥

[pallid falcon]

Here’s my work

[pallid falcon]

Is this problem hard then?

The graph shown in the final answer has the interval [-2π, -π] positive, but why if it’s negative (decreasing) on that same interval, as I showed in my work.

[pallid falcon]

How?

[crimson locust]

how did you show it was decreasing

because part of it is decreasing

the other part is increasing

[pallid falcon]

By plugging in values into f’

[crimson locust]

did you plug -2π into f'

[pallid falcon]

Are there other roots, besides pi, -pi and 0?

[pallid falcon]

No

[crimson locust]

-2π and 2π are solutions

since all solutions of sin(x) are solutions of x^2sin(x)

[pallid falcon]

What does solutions mean?

[crimson locust]

roots

[pallid falcon]

Yes. But 2pi and -2pi are the endpoints

[crimson locust]

and?

they're still on the x-axis

[pallid falcon]

But how do they determine if the graph is increasing or decreasing if they’re endpoints?

[crimson locust]

x^2sin(x) is literally positive whenever sin(x) is positive and nowhere else

it would genuinely be easier to just think of this graph as sin(x) multiplied by x^2 instead of using derivatives for everything

[pallid falcon]

What? I’m trying to plot the graph using first and second derivative tests, which is what I learned in the section of the textbook I’m on

[crimson locust]

then i wish you luck

[pallid falcon]

So does using derivatives work in this case?

[pallid falcon]

What do you mean? Can you explain?