Answer to Question #104860 in Calculus for Kassim

Question #104860
If k ≥ 1, the graphs of y = sinx and y = keex
intersect for
x ≥ 0. Find the smallest value of k for which the graphs are tan￾gent.
k =
What are the coordinates of the point of tangency?
x = , y = .
1
Expert's answer
2020-03-16T11:03:54-0400

For tangency,

sinx=kexsin x=ke^{-x} and cosx=kexcos x= -ke^{-x}

sinx=cosx=kexsinx=-cos x=ke^{-x}

tanx=1forx=nπ+3π/4tanx=-1 for x= nπ+3π/4

1/2=ke3π/4;x=3π/4;n=01/\sqrt{2}=ke^{-3π/4} ; x = 3π/4 ; n=0

k=e3π/42k=\frac {e^{3π/4}}{\sqrt{2}}

Coordinates of the point of tangency (3π4,12)(\frac{3π}{4},\frac 1{\sqrt{2}})


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