Finding the Zeros of f(x)=e^xsin(x) on [0,2π]

[UrbanXrisis]

I'm not quite sure what the question is asking:
"If $$f(x)=e^xsin(x)$$, then the number of zeros of $$f$$ on the interval [o,2pi] is?"

I took the derivative of this and found where it was equal to zero:
$$f(x)=e^xsin(x)$$
$$f'(x)=e^xsin(x)+e^xcos(x)$$
$$0=sin(x)+cos(x)$$

I got zero. However, tha answer is 3, any suggestions?

 

[Nylex]

No, it wants where the function is zero, not where the gradient of the curve is zero. Where is sin x zero in that interval?

 

[UrbanXrisis]

you mean when $$0=e^xsin(x)$$?

 

[dextercioby]

I'm not quite sure what the question is asking:
"If $$f(x)=e^xsin(x)$$, then the number of zeros of $$f$$ on the interval [o,2pi] is?"

I took the derivative of this and found where it was equal to zero:
$$f(x)=e^xsin(x)$$
$$f'(x)=e^xsin(x)+e^xcos(x)$$
$$0=sin(x)+cos(x)$$

I got zero. However, tha answer is 3, any suggestions?

$$e^{x}\sin x=0\Rightarrow \sin x=0$$
Solve the last equation on the interval $[0,2\pi]$

Daniel.

 

[Nylex]

you mean when $$0=e^xsin(x)$$?

Yes, yes I do (sorry had to lengthen my post).

 

[UrbanXrisis]

it's is zero when x=0, x=pi, x=2pi

so it hits three times!

thanks! Also, why doesn't the e^x make a difference?

 

[dextercioby]

it's is zero when x=0, x=pi, x=2pi
so it hits three times!
thanks! Also, why doesn't the e^x make a difference?

It never annulates.Not even for complex arguments.

Daniel.

EDIT:Cause it never annulates,it does not affect the zero-s of the function.Plot the graph of 'f'.U'll see quite an interesting behavior.It has no limit for x->+infty.At minus infty it goes to zero.

 

[UrbanXrisis]

so e^x only increases the sinX amplitude, never now far it streatchs, so it doesn't effect how many times sinX crosses the x-axis

 

[HallsofIvy]

"Annulates"? I assume you mean "is never equal to 0" but the only definition I can find of "annulate" is "ring shaped".

UrbanXrises: It's not so much that it is 'always increasing'. In order to solve AB= 0, you solve A= 0 and B= 0. Since e^x is never 0, The only solutions of e^xsin(x)= 0 are where sin(x)= 0.