1 | initial version |

Those two roots are both approximations of 0. If possible, you should always seek to solve theoretically and numerically. If the answers agree then you feel confident everything was done correctly. If the answers disagree then you have to look deeper. The roots of the function are when it is equal to 0, so you need to solve the equation `(x^2)*cos(2*x)=0`

. That means either `x=0`

or `cos(2*x)=0`

. Now cos(x)=0 at +/- (2k+1)*pi/2 so cos(2x)=0 at +/- (2k+1)*pi/4. Since we're looking for roots over [-10,10], this means our roots are 0, +/- pi/4, 3

2 | No.2 Revision |

Those two roots are both approximations of 0. If possible, you should always seek to solve theoretically and numerically. If the answers agree then you feel confident everything was done correctly. If the answers disagree then you have to look deeper. The roots of the function are when it is equal to 0, so you need to solve the equation `(x^2)*cos(2*x)=0`

. That means either `x=0`

or `cos(2*x)=0`

. Now cos(x)=0 at +/- (2k+1)*pi/2 so cos(2x)=0 at +/- (2k+1)*pi/4. Since we're looking for roots over [-10,10], this means our roots are 0, +/- pi/4, 3

You mention inflection points but inflection points involve the second derivative. The answers you have are from setting the first derivative equal to 0. If you rewrite the second answer as `x=cot(2x)`

and think about the graphs of cot(2x) and x and where the intersect you'll get an idea of the number of answers. There is no way to solve that equation analytically.

3 | No.3 Revision |

`(x^2)*cos(2*x)=0`

. That means either `x=0`

or `cos(2*x)=0`

. Now cos(x)=0 at +/- (2k+1)*pi/2 so cos(2x)=0 at +/- (2k+1)*pi/4. Since we're looking for roots over [-10,10], this means our roots are 0, +/- pi/4, 3

You mention inflection points but inflection points involve the second derivative. The answers you have are from setting the first derivative equal to 0. If you rewrite the second answer as `x=cot(2x)`

and think about the graphs of cot(2x) and x and where the intersect you'll get an idea of the number of answers. There is no way to solve that equation ~~analytically.~~analytically. However, since you know where, approximately, the roots are you can use `find_root`

, documentation here to find them. Your first positive solution would be found by, for example, `find_root(cot(2*x)-x,.01,.8)`

to get `0.5384369931559019`

.

4 | No.4 Revision |

`(x^2)*cos(2*x)=0`

. That means either `x=0`

or `cos(2*x)=0`

. Now cos(x)=0 at +/- (2k+1)*pi/2 so cos(2x)=0 at +/- (2k+1)*pi/4. Since we're looking for roots over [-10,10], this means our roots are 0, +/- pi/4, 3

You mention inflection points but inflection points involve the second derivative. The answers you have are from setting the first derivative equal to 0. If you rewrite the second answer as `x=cot(2x)`

and think about the graphs of cot(2x) and x and where the intersect you'll get an idea of the number of answers. There is no way to solve that equation analytically. However, since you know where, approximately, the roots are you can use `find_root`

, documentation here to find them. Your first positive solution would be found by, for example, `find_root(cot(2*x)-x,.01,.8)`

to get `0.5384369931559019`

~~.~~. What should become apparent is that mathematical theory can help you use SAGE like an expert. Just learning to use SAGE is not enough.

5 | No.5 Revision |

`(x^2)*cos(2*x)=0`

. That means either `x=0`

or `cos(2*x)=0`

. Now cos(x)=0 at +/- (2k+1)*pi/2 so cos(2x)=0 at +/- (2k+1)*pi/4. Since we're looking for roots over [-10,10], this means our roots are 0, +/- pi/4, 3

You mention inflection points but usually we calculate inflection points ~~involve ~~by taking the second ~~derivative. The ~~derivative but from your answers ~~you have are from ~~I see you are setting the first derivative equal to 0. Where the derivative is 0 (or undefined) are usually called the critical values. If you rewrite the second answer as `x=cot(2x)`

and think about the graphs of cot(2x) and x and where the intersect you'll get an idea of the number of answers. There is no way to solve that equation analytically. However, since you know where, approximately, the roots are you can use `find_root`

, documentation here to find them. Your first positive solution would be found by, for example, `find_root(cot(2*x)-x,.01,.8)`

to get `0.5384369931559019`

. What should become apparent is that learning mathematical theory can help you to use SAGE like an ~~expert. Just learning to use SAGE is not enough.~~expert.

6 | No.6 Revision |

`(x^2)*cos(2*x)=0`

. That means either `x=0`

or `cos(2*x)=0`

. Now cos(x)=0 at +/- (2k+1)*pi/2 so cos(2x)=0 at +/- (2k+1)*pi/4. Since we're looking for roots over [-10,10], this means our roots are 0, +/- pi/4, 3

You mention inflection points but usually we calculate inflection points by taking the second derivative but from your answers I see you are setting the first derivative equal to 0. Where the derivative is 0 (or undefined) are usually called the critical values. If you rewrite the second answer as `x=cot(2x)`

and think about the graphs of cot(2x) and x and where the intersect you'll get an idea of the number of answers. Try plotting

```
A = plot(cot(2*x),-10,10,ymax=10, ymin=-10, detect_poles=True)
B = plot(x,-10,10,ymax=10, ymin=-10,color='red')
(A+B).show()
```

Which looks like:

There is no way to solve that equation analytically. However, since you know where, approximately, the roots are you can use `find_root`

, documentation here to find them. Your first positive solution would be found by, for example, `find_root(cot(2*x)-x,.01,.8)`

to get `0.5384369931559019`

. What should become apparent is that learning mathematical theory can help you to use SAGE like an expert.

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