Forwhatvaluesofcisthereastraightlinethatintersectsthecurvey=x^4+cx^3+12x^2-5x+2infourdistinctpoints?大概意思是,c为何值时,一条直线可以与曲线y=x^4+cx^3+12x^2-5x+2相交有四个不同点?

Question

For what values of c is there a straight line that intersects the curve y=x^4 + cx^3+12x^2-5x+2 in four distinct points?
大概意思是,c为何值时,一条直线可以与曲线y=x^4 + cx^3+12x^2-5x+2相交有四个不同点?

Answer 1

依题意 则y'应该至少有3个零点
y'=4x^3+3cx^2+24x-5
发现最多有3个0点
故y只能有3个零点
令y'=0
则4x^3+3cx^2+24x-5=0应该有个3不等实根
然后得分解因式 得有3个因式 每个因式=0 求出c就行...