C2/C3/C4? Calculus Please Help????!!!!

I've been stuck on this particular question for two days now:

f(x)=e^2x sin2x
a) Use calculus to find the coordinates of the turning points on the graph of y=f(x).

b) Show that f''(x)=8e^2x cos2x

c) Hence, or otherwise, determine which turning point is a maximum and which is a minimum.

I've only managed part of a )and only a little of b) Here's my working so far:

a) f'(x)= e^2x times 2cos2x + 2e^2x times sin2x
f'(x)= 2e^2x(cos2x) + 2e^2x(sin2x)
2e^2x( sin2x + cos2x )=0
2e^2x=0 or sin2x + cos2x=0
sin2x = -cos2x So divide both sides by cos2x
So tan2x = -1. That's all I've been able to do so far for a), I have no idea what to do next as the particular book I'm woking on only explains the derivates and intergration of sin and cos but not tan. I've understood sin/cos = tan because of a previous question also resulting in tan and connecting the dots. But I have absolutely no idea what to do from there, I just know sin/cos=tan.

In regards to b): f'(x)= 2e^2x( sin2x + cos2x )
So f''(x)= 2e^2x times ( 2cos2x -2sin2x ) + 4e^2x times ( sin2x + cos2x ). I have tried to simplify from there but it's no use since I have no idea how I'm supposed to prove the answer is 8e^2x cos2x.

I haven't even attempted to start c) yet.

Thank you to anyone who took the time to read this and can enlighten me. Calculus can sometimes be very confusing/irritating!

So tan2x = -1. That's all I've been able to do so far for a), I have no idea what to do next as the particular book I'm woking on only explains the derivates and intergration of sin and cos but not tan. I've understood sin/cos = tan because of a previous question also resulting in tan and connecting the dots. But I have absolutely no idea what to do from there, I just know sin/cos=tan.

For a) you need to now solve tan2x = -1. Do you have any ideas for this?

Also, I assume there's a restriction on x that you didn't include in the question e.g. 0<x<2pi ?

You don't need to differentiate here because tan(2x) = -1 is not really a function of x any more (just a constant) - some x satisfies this equation. Your f''(x) is correct and expanding the brackets gives you the required result.

No, unfortunately I have absolutely no idea. And yes you are 100% right, thank you for noticing that I just assumed it had no importance, it's: x≤x≤pi. I will edit in my question now.

Have you never solved trig equations before? E.g. something like this:

$\cos x = 0.2$ ?

Students generally learn this stuff before tackling product rule differentiation. Are you self-studying?

No, I don't think I've solved something like that before, and yes I'm self studying.

Edit: Trigonometric equations like the one you've mentioned are in the following chapter to the current Calculus one I'm on. Seems kind of stupid to me now since I need some sort of understanding of them to solve the calculus that I am doing.

How are you self-studying? Are you reading textbooks from the British A Level syllabus e.g. C1/2 etc.

If you need any self-studying advice or have other questions then please continue to post in this forum. I self-studied maths with the help of TSR and plenty of others have too

Thank you very much for your welcoming onto the forum. I think that this is the first time I've ever posted a question anywhere, when the answers don't have any working out, I usually either try and search online or spend as much time as it needs, sometimes even up to a week. It's also good to know that you've self studied as well. And yes, I'm reading British A Level textbooks. Could you help me with the tan2x = -1 problem?

Edexcel? Product rule differentiation is in C3 and trig equations like this are in C2. So I recommend going back and learning this topic if you want to do this question.

It will be hard to help you with this question if you haven't learnt the topic before.

The questions in the Trig chapter consist of: converting angles in radians to degrees, converting angles to degrees, using a calculator to find values of Trig functions, converting angles to radians and stuff of that sort. And then it delves into shapes. Can't really see anything similar to the problem I'm facing.

Is this Edexcel C2?

Later on in C2 there will be a chapter with a name like "Trigonometric identites and equations".

I've found these videos online explaining trig equations:
https://www.youtube.com/watch?v=OLzXqIqZZz0
https://www.youtube.com/watch?v=ZSVnNhKbNUg
Thanks for your help.

Those are great but you'll also need to understand the CAST diagram and how to use it to solve trig equations.

I recommend these videos:

https://www.youtube.com/watch?v=tFBQ2YMdfhU
https://www.youtube.com/watch?v=zGCnBgyM9kQ

This is quite a major topic by the way so you'll need to spend some time watching videos and trying lots of questions.