Question 80891This question is from textbook CALCULUS
: For the general cubic polynomial f(x)= ax^3 + bx^2 + cx + d (a different from 0).
Find conditions on a, b, c, and d to ensure that f is always increasing or always decreasing on (-00, +00).
This question is from textbook CALCULUS
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! For the general cubic polynomial f(x)= ax^3 + bx^2 + cx + d (a different from 0).
Find conditions on a, b, c, and d to ensure that f is always increasing or always decreasing on (-00, +00).
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To assure Increasing:
The slope would have to be postive for all values of x.
Take the derivative: 3ax^2+2bx+c>0
As long as c>0 and a>0 you would get a graph (parabola) that
is always above the x axis. That should meet the asked-for
condition.
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To assure decreasing:
The slope would have to be negative for all values of x.
Use the same drivative but make it <0
Let c<0 and a<0 then you would get a graph (parabola) that
is always below the x-axis. That should meet the asked-for
condition.
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Cheers,
Stan H.
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Cheers,
Stan H.
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