Finding value of k so two curves are tangent

AI Thread Summary
To find the value of k for which the curves y=sin x and y=ke^{-x} are tangent, two conditions must be satisfied: f(x_0)=g(x_0) and f'(x_0)=g'(x_0). By equating the derivative of the two equations, cos x = -ke^{-k}, a system of equations is established with unknowns x_0 and k. The trigonometric nature of the equations indicates multiple possible values for x_0, leading to corresponding values for k(x_0). Ultimately, there will be a minimum value for k that satisfies the tangency condition.
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Homework Statement
Find the smallest value of k (where k ≥ 1) so that y = sin x and ##y=ke^{-x}## are tangent for x ≥ 0
Relevant Equations
Derivative
I tried to equate the derivative of the two equations:
$$\cos x=-ke^{-k}$$

Then how to continue? Is this question can be solved?

Thanks
 
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In order for two curves ##y=f(x),y=g(x)## to be tangent at a point ##x_0## two conditions should hold:
  1. ##f(x_0)=g(x_0)##
  2. ##f'(x_0)=g'(x_0)##
So put these two conditions into work for the specific ##f(x)=\sin x## and ##g(x)=ke^{-x}##. You ll get a system of two equations with two unknowns ##x_0## and ##k##. It will turn out that you will have many possible values for ##x_0## (trigonometric equation involved) and from the corresponding values for ##k(x_0)## there will be a minimum.
 
Thank you very much Delta2
 
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