Use process of optimization

Do you mean f(x) = e^{-x} sin(2x)?

f'(x) = e^{-x} (2cos(2x) - sin(2x))

Solve f'(x) = 0 for extreme values,

tan(2x) = 1/2

x = .2318 rad , f(.2318) = .3547 <==Answer

Use process of optimization

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Do you mean f(x) = e^{-x} sin(2x)?

f'(x) = e^{-x} (2cos(2x) - sin(2x))

Solve f'(x) = 0 for extreme values,

tan(2x) = 1/2

x = .2318 rad , f(.2318) = .3547 <==Answer

Grigori S. | Certified Physics and Math Teacher G.S.Certified Physics and Math Teacher G.S.

The way they write equations and formulas sometimes is incorrect. You can write the same function in the way as Robert J. did, or as f(x) = exp(-xsin2x). If the exp(-x) is to be multiplied by sin(2x) then it is to be included in parenthesis, such as f(x) = exp(-x) sin(2x) , which is a totally different function. If we take

f(x) = exp( -xsin(2x)) then f'(x) = 0 if tan(2x) =-2x. This equation can be satisfied if x = 0 which is visible

without differentiation. If f(x) = exp(-x) sin (2x) then f'(x) = exp(-x)(2cos(2x) -sin(2x)) = 0 or

tan(2x) = 2 x = (1/2) atan(2) (cos(2x) = 1 and can be a divisor)

Because tan (2x) is a periodic function with the period = pi, we can write more common solution in the following way: x = (1/2)(atan(2x) +(or -) pin)

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## Comments

it is e minus x

Thank you so much, Robert! Can you, please, also show how did you find x.