Hello,
They'll have the same (at least one) tangent if there are more than one solution a to the equation
Obviously if there is a common tangent that means that there has to be a point on each curve where the gradient is the same.
So do as moo suggested and solve .
Now determine the point on each curve that has this x-coordinate. Now calculate the tangent at this point for each curve. Do you get the same line?
Now, for , if we here take , then
and the equation of the tangent becomes
The equation of the tangent to the parabola becomes:
Compare these two equations. You'll get a system of equations:
Solve for a and b.
Remark I: If you have drawn a rough sketch of both graphs you probably would have noticed that there must be at least one common tangent.
Remark II: I haven't found a solution of this system - but if I succeed I'll be back.
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Here I am again:
I substituted in the second equation. I wasn't able to solve this equation algebraically so I used Newtons method and got an approximate result:
I've attached the graphs of the three functions.