f(x)=alltermsf
.0
and 0
into the Answer boxes.
xmin < x <
xmax
Differentiating, we obtain:
f'(x)=alltermsfpr=factoredfpr
Factor f'(x)
, if possible.
The factorization is given by f'(x)=3*p(x+2*a)(x-2*b)
.
Using this factorization,
figure out where
the derivative f'(x)
is positive and negative.
Notice that f'(x)
is negative when x
is a large positive number.
Work along the number line from right to left.
The derivative f'(x)
is
negative on xmax < x
,
positive on xmin < x < xmax
and
negative on x < xmin
.
Then the function f(x)
is
decreasing on xmax < x
,
increasing on xmin < x < xmax
and
decreasing on x < xmin
.
Now enter xmin
and xmax
into the Answer boxes,
to indicate that xmin < x < xmax
is the largest (bounded) open interval of increase.
NOTE: The next two hints show: the graph of f
, followed by the graph of f'
.
Note that the graph of f
runs uphill between x = xmin
and x = xmax
.
Note that the graph of f'
is above the x
-axis between x = xmin
and x = xmax
.
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] ); });