## anonymous 3 years ago What is the maximun of f(x)=3cosx+2sinx ? Whithout using derivatives

1. anonymous

2. anonymous

@seidi.yamauti are you aware of what the graphs of cosx and sinx look like?

3. anonymous

@abayomi12 he said without using derivatives.

4. anonymous

ok

5. anonymous

I'd simplify this first: f(x) = 2 * sin(x) * cos(x) f(x) = sin(2x)

6. anonymous

max value of acosx+bsinx is sqrt(a^2+b^2) and min is -sqrt(a^2+b^2) thus here max value is sqrt(3^2+2^2)

7. anonymous

$a\cos x+b\sin x=\sqrt{a^2+b^2} (\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\sqrt{a^2+b^2}}\sin x)$ using the fact that$(\frac{a}{\sqrt{a^2+b^2}})^2+(\frac{b}{\sqrt{a^2+b^2}})^2=1$ u can prove$\sqrt{a^2+b^2} (\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\sqrt{a^2+b^2}}\sin x)=\sqrt{a^2+b^2} \sin(x+\alpha)$or$\sqrt{a^2+b^2} (\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\sqrt{a^2+b^2}}\sin x)=\sqrt{a^2+b^2} \cos(x+\beta)$

8. shubhamsrg

cool.. B|

9. phi

in other words, use the identity sin(x+y) = sin(x)cos(y)+cos(x)sin(y)

10. anonymous

maximum = sqrt(3^2 + 2^2) = sqrt(13)

11. anonymous

अगर आप चाहें तो वोल्फ्रम का उपयोग कर सकते हैं.

12. anonymous

बात समझ आई?

13. anonymous

@mukushla आप तो मास्टर हो, मास्टर!

14. anonymous

man this is hindi !

15. anonymous

ye cid walay yahan kya kar rahay hain :P

16. anonymous

@sami-21 Kya chal raha hai?

17. anonymous

@sami-21 chutti pe hain hum sab

18. anonymous

humari police

19. anonymous

btw just to let you know guys irreverent answers are considered as SPAM .

20. anonymous

lol

21. anonymous

You can use a special trig identity: Acos(x) + Bsin(x) = Ccos(x-y), where C=\sqrt{A^2=B^2) and y = arctan(B/A). Hence $f(x)=3\cos x + 2\sin x = \sqrt{13}\cos\left(x-arctan\tfrac{2}{3}\right)\approx\sqrt{13}\cos\left(x-.588\right)$ But in any case, the maximum for cos is 1, so the maximum for f(x) is the square root of 13.

22. anonymous

also $$a\sin(x)+b\cos(x)=\sqrt{a^2+b^2}\sin(x+\theta)$$ for suitable $$\theta$$ as i recall you can get this from "addition angle" formula

23. anonymous

@satellite73, how are you inserting equations without linebreaks? $signs don't seems to work here (as they would in LaTeX). 24. anonymous use \( instead of \[ 25. anonymous Thank you. 26. anonymous you are right,$ does not work here turns out \( is an alternative in latex if you need to see any code, right click and you can see it. it is good method for copying and pasting as well, so you don't have to rewrite the latex every time