How to check weather 2 graphs touch or intersection each other

Question

Till date what i did was to find their intersection point If it exists and If it does i would find slope of tangent at that point and If both have same slope i would say they touch and If different i would say they intersect but it failed when i tried for $x^3$ and $x^5$ at X=0 according to my method they touch each other but actually they intersect at X=0 . After that the only method which came in my mind was to draw graph . But it is not possible for every polynomial function to draw graph manually. Is there any other method

Answer 1

When you found the common point of $f(x)$ and $g(x)$. Find $f(x) - g(x)$ just before and after the common point. If it changes sign, the two curves definitely intersect, if it does not change sign then they definitely do not intersect. Make sure the range you are taking to check holds no other common points. For the reason why your method does not work, please check this.

Answer 2

The important thing is the order of the first derivative where they disagree. If it is odd, they cross and if it is even, they touch. You can imagine expanding the difference of the two functions in a Taylor series. The fact that they have a common point says the constant term of the series is zero. If the first derivatives differ, the Taylor series will have a term in $x$, which is positive on one side of the intersection and negative on the other side, so they cross. If the first derivatives agree but the second derivatives disagree, the leading term in the Taylor series will be $x^2$, which has the same sign on both sides of the touching point. In your example, the Taylor series of the difference is $x^3-x^5$ which has the first nonzero derivative as the third. Because it is an odd term, the functions cross.

Answer 3

Set the polynomials equal, and solve for relations amongst variables. In the case of univariate same variable polynomials this means solving for x usually. ex. when does $$10x+2=x^2+12x+3$$ turns out, at the roots( zero points) of $$x^2+2x+1=(x+1)^2$$, that means x=-1 (counted twice because of multiplicities).