It is given that the graph of y= x⁴ + ax³ + bx² + c (where a , b , c , d are real) cuts the x axis at least 3 distinct points. Then show that it either cuts the x axis at 4 distinct point or 3 distinct point and at any one of these three points we have a maxima or minima.
when a four degree equation cuts the x axis at 3 points (provided that all the roots are real) it means one of these three roots have a multiplicity of 2. hence note that for the function :
f(x)=(x-a)^2(x-b)(x-c); f(a)= 0 also: f"(a)=0 Implies maxima or minima.... hence proved.
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