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

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?

1 Answer
Jun 10, 2018

Please see below.

Explanation:

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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.