Challenge Question

[schoolstudent115]

Guys I want to give a question I made (I’ve made a worked solution), and see how you guys solve it.

Without using a calculator:
Let f(x)=mx, where m>0, g(x)=e^x
a) Find the value of m for which f(x)=g(x) has exactly one solution.
b) Hence, find the coordinates of this point of intersection.

[Harrycc3000]

as there is only one solution, y=mx is a tangent of g(x)
therefore, gradient (m) is equal to g'(x)
therefore m=e^x
the intersection between this tangent and g(x) can be found by solving f(x)=g(x),
therefore for this tangent, mx=e^x
as m=e^x and mx=e^x,
m=mx
therefore
x=1 (x value at which gradient and y value is equal (which is x value of the intersection between e^x and its tangent))
therefore m(1)= e^(1)
therefore m=e
tangent is y=ex and point of intersection is at (1,e)