# Thread: two functions ( common tangent in a common point )

1. ## two functions ( common tangent in a common point )

The functions from the picture have a common tangent in a common point if:
A. a=1+e
B. a=0
C. a=1
D. a=e-pi
E. a=-1

I know that the conditions are: f(x)=g(x) and f'(x)=g'(x)
I tried to solve the system but I did't get too far.

2. ## Re: two functions ( common tangent in a common point )

Originally Posted by Vali
The functions from the picture have a common tangent in a common point if:
A. a=1+e
B. a=0
C. a=1
D. a=e-pi
E. a=-1
I know that the conditions are: f(x)=g(x) and f'(x)=g'(x)
I tried to solve the system but I did't get too far.
The functions are: $f(x)=x+\sqrt{x^2+a},~a\ge 0~\&~g(x)=x^2+1$ then $f'(x)=1+\dfrac{x}{\sqrt{x^2+a}}~\&~g'(x)=2x$

I submit to you that there is a valid answer in that list. Look at the graph

3. ## Re: two functions ( common tangent in a common point )

The algebra involved is pretty much undoable by hand.

What I suggest is that for each of those values of $a$ listed you plot $f$ and $g$ and see which value makes the two intersect at a tangent point.
DESMOS, or some other software will help with this.

Once you can visualize the situation the correct answer is pretty simple to see.