randRange(-2,-1) randRange(-4,4) randRange(-4,4) randRange(1,100) a>0?"("+(-2)*a+")":(-2)*a -3*a*p 3*p*(a*a+b*b) -2*a a*a+b*b q===0?"":(q>0?"+"+q+"x^2":"-"+(-q)+"x^2") p===-1?"-x^3":p+"x^3" cutermf+qutermf+"-"+(-r)+"x+"+s q===0?"":(q<0?"-"+(-2*q)+"x":"+"+(2*q)+"x") (3*p)+"x^2"+lintermfpr+"-"+(-r) q===0?"":(q<0?"+"+(twoqover3p)+"x":"-"+(-twoqover3p)+"x") "x^2"+lintermfprover3p+"+"+rover3p 3*p+"("+alltermsfprover3p+")" 10 -10 p*xmin*xmin*xmin+q*xmin*xmin+r*xmin+s p*xmax*xmax*xmax+q*xmax*xmax+r*xmax+s 1000*floor((ymin-20)/1000) 1000*ceil((ymax+20)/1000) [ -10, 10 ] [ flooredymin, ceiledymax ]
This is an interval of increase problem.
Find the largest (bounded) open interval of increase for
      f(x)=alltermsf.
If there are no intervals of increase,
      or if every maximal interval of increase is unbounded,
      then enter 0 and 0 into the Answer boxes.

0 < x < 0

Differentiating, we obtain:
       f'(x)=alltermsfpr=factoredfpr
Factor f'(x), if possible.

The discriminant of alltermsfprover3p is
       minustwoa^2-4(1)((a*a+b*b)) =4*a*a-4*(a*a+b*b),
   which is negative, so f'(x) does not factor (over the real numbers),
   so f'(x) has no (real) roots.

The derivative f'(x) is negative for large positive numbers x.

Thus, the derivative f'(x) is negative everywhere.

Then the function f(x) is decreasing everywhere,
   and so f(x) has no intervals of increase.

Following the directions in the problem, enter 0 and 0 into the Answer boxes,    to indicate that there are no intervals of increase.

NOTE: The next two hints show: the graph of f, followed by the graph of f'.
Note that the graph of f always runs downhill.
Note that the graph of f' is always below the x-axis.

initAutoscaledGraph( [ xrange , yrange ], {} ); style({ stroke: "#6495ED", strokeWidth: 3 }, function() { plot( function( x ) { return p*x*x*x+q*x*x+r*x+s; }, [-10,10] ); });

initAutoscaledGraph( [ [ -10, 10] , [ -500, 500] ], {} ); style({ stroke: "#6495ED", strokeWidth: 3 }, function() { plot( function( x ) { return 3*p*x*x+2*q*x+r; }, [-10,10] ); });