Question

It is given that the graph of $y=x^4+a{x}^3+b{x}^2+cx+d$ (where $a,b,c,d$ are real) has at least 3 points of intersection with the x-axis. Prove that either there are exactly 4 distinct points of intersection, or one of those 3 points of intersection is a local minimum or maximum.

Answer 1

$y=x^4+a{x}^3+b{x}^2+cx+d$ ( at least $3$ points of intersection with x-axis)

Case $1$: One point of intersection

$(x-\alpha )(k{x}^3+\beta x^2+\alpha x+\delta )=0$

Then $(k{x}^3+\beta x^2+\alpha x+\delta )$ won't have solution.

But cubic polynomial will yield one more real solution

Case $2$: Three points of intersection

$(x-\alpha )(x-\beta )(x-\delta )(mx+h)=0$

Now, if there are $3$ points of intersection

There must be one point which is either local maximum or minimum. As if there are $3$ solutions, there will be a compulsory fourth solution.