Mikey's question at Yahoo! Answers regarding finding the tangent point of two functions

MarkFL

Staff member
Here is the question:

Find the point where the graphs of f(x)=x^3-2x and g(x)=0.5x^2-1.5
are tangent to each other; or have a common tangent line.

I'm pretty sure I have to find the derivatives of each and set them equal to eachother. Do I then just solve for x? My answer is seems weird. Can you explain how you would go about this problem please?

Thanks.
I have posted a link there to this topic so the OP can see my work.

MarkFL

Staff member
Hello Mikey,

If the two given functions are tangent to each other, then the difference between the two functions will have a repeated root:

$$\displaystyle f(x)-g(x)=(x-a)^2(x-b)$$

$$\displaystyle x^3-\frac{1}{2}x^2-2x+\frac{3}{2}=x^3-(2a+b)x^2+\left(a^2+2ab \right)x-a^2b$$

Equating corresponding coefficients, we find:

$$\displaystyle 2a+b=\frac{1}{2}$$

$$\displaystyle a^2+2ab=-2$$

$$\displaystyle a^2b=-\frac{3}{2}$$

The third equation implies:

$$\displaystyle b=-\frac{3}{2a^2}$$

Using this, the first equation gives:

$$\displaystyle 4a^3-a^2-3=0\implies a=1,\,b=-\frac{3}{2}$$

And so the two functions are tangent to one another at $x=1$. Here is a plot: