Question

Please Help.
Find all relative maxima and minima for the function.

Answer 1

I prefer to write as
f(x)=e^x*cos(2x)

Answer 2

then find the derivative

Answer 3

I found the derivative. I do not know what to do next

Answer 4

Let me see your derivative

Answer 5

\[e ^{x}(\cos(2x)-2\sin(2x)\]

Answer 6

ok so we know e^x is never 0

Answer 7

we need to find when \[\cos(2x)-2\sin(2x)=0\]

Answer 8

I solved for x but I still don't know how to express the max and min. I got x = \[\frac{ 1 }{ 2 } \arctan(\frac{ 1 }{ 2 })\]

Answer 9

are we looking at a certain interval?

Answer 10

No, unfortunately

Answer 11

\[\cos(2x)=2 \sin(2x) \\ \frac{1}{2}=\tan(2x) \\ \frac{1}{2}=\tan(2x+\pi \cdot n) \text{ since \tan has period \pi } \\ \arctan(\frac{1}{2})=2x+ \pi \cdot n\]
But I'm not tottally positive this gives us all of our critical numbers when solving for x

Answer 12

but anyways to figure out what the rel max and rel min values are ... we need to plug back into the function
but we also need to determine for which n , we have a max occurring at x
and for which n we have a min occurring at x
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unfortunately I have to go now
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Answer 13

Please continue. I really need to understand this

Answer 14

Never mind. I got it. I just had to look at a different way. Thanks for your help.