Graphs

HOW TO PLOT THE GRAPH OF CUBIC POLYNOMIALS?

LET  US  consider  a  general  3  degree  equation   F(x) -- ax³+bx²+cx+d , a >0

STEP -1) find f '(x) i.e.  3ax²+2bx+c,  if dicriminant of thif eqn. = D {=(2b)²-4(3a)(c) < o }, then only one real solution . and graph (of  F(x) ) is --- always inc.  from  -infinity to +infinit and cuitting x-axis at exactly one point .

STEP-2) if D =0 , then 3 real equal ruts . again cutting x-axis at only one point and graph ( of F(x) }  is from -infinity to +infinity. ( always increasing )

STEP-3) if D > 0 , then let two solutions of f ' (x)  be  a₁ &  a₂ .  SO, F(a₁)  will be minimum  and  F(a₂) will be maximum. and so, the graph of  f(x) will increase 4m  x= - infinity  to x= a₁ and decreases 4m x= a₁ to x-a_{2 .} again graph increases from x=a₂ to  =infinity