If k is greater than or equal to 1, the graphs of y=sinx and y=k(e^(-x)) intersect for x is greater than or equal to 0.
A) Find the smallest value of k for which the graphs are tangent.
B) What are the coordinates of the point of tangency?
Given the functions `f(x)=ke^(-x),k>1` and `g(x)=sinx` we want to find the smallest `k` such that the graphs are tangent.
(1) For the graphs to intersect you need `f(x)=g(x)` :
(2) For the graphs to be tangent, they must share a tangent line. Thus the first derivative, which gives the slope of the tangent line, must be the same at some point or `f'(c)=g'(c)` .
(3) The derivatives:
(4) Now we have a set of simultaneous equations. At the point of intersection `f(x)=g(x)` , and if the graphs are tangent at that point then `f'(x)=g'(x)` . So:
`=>x=(3pi)/4+npi,n in ZZ`
(5) Let `x=(3pi)/4` . The slope of the tangent line for both curves is `m=(-sqrt(2))/2` and the shared point is `((3pi)/4,(sqrt(2))/2)`
At this point `k=sqrt(2)/2e^((3pi)/4)`
The straight line shown is the tangent line: