How to prove that the graph of the functions are tangent to eachother.? Also determine the point of tangency? Thank you

Question

Function f with function rule f(x) = 5 - 2x
Function g with function rule g(x) = #sqrt(5-x²#

I know that if you derive the functions, you get the tangents. But then, I'm stuck! Can you just equate them to eachoter?

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Answer 1 · 2018-06-10T17:10:11

Please see below.

.

#f(x)=5-2x#

#g(x)=sqrt(5-x^2)#

If they are tangent to each other they have a common point which is the point of tangency. Let's set them equal to each other and find the point:

#sqrt(5-x^2)=5-2x#

#5-x^2=(5-2x)^2#

#5-x^2=25-20x+4x^2#

#5x^2-20x+20=0#

#5(x^2-4x+4)=0#

#x^2-4x+4=0#

#(x-2)^2=0#

#x=2#

Let's find the slope of the tangent line to g(x) at #x=2#:

#m=dy/dx=1/2(5-x^2)^(-1/2)(-2x)=(-x)/sqrt(5-x^2)=(-2)/1=-2#

As you can see, #f(x)# is a straight line with a slope of #-2# because the general form of the equation of a straight line is:

#y=mx+b#

Comparing this with #f(x)# gives us #m=-2#

Therefore, #f(x)# is the tangent line to #g(x)# at #x=-2#.