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y' is increasing

The values of x for which y= 2x3 - 12x2 +18x +7=0 is increasing are?

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Anthony B. | Math, Stats, Calculus, APA TutorMath, Stats, Calculus, APA Tutor
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Hello Tiarne,
 
To find where this function is increasing, we have to find where the derivative in negative.
y= 2x3 - 12x2 +18x +7
y'=6x2-24x+18
 
We need to find the x intercepts, so set equal to 0.
0=6x2-24x+18
0=6(x2-4x+3)
0=6(x-3)(x-1)
x=3 or x=1
 
When y' is positive, this means that the original function has a positive slope and is increasing.  Since y' is a parabola that opens upward, it is negative only for values between 1 and 3, and positive outside of these values.
 
So we can say that y is increasing at:
-∞<x<1  and 3<x<∞
and y is decreasing at:
1<x<3