Answer:
a) The increasing intervals would be from negative infinity to -6 and 3 to infinity. The decreasing interval would just be from -6 to 3
b) The local maximum comes at x = -6. The local minimum would be x = 3
c) The inflection point is x= -3/2
Step-by-step explanation:
To find the intervals of increasing and decreasing, we can start by finding the answers to part b, which is to find the local maximums and minimums. We do this by taking the derivatives of the equation.
f(x) = 4x^3 + 18x^2 - 216x + 3
f'(x) = 12x^2 + 36x -216
Now we take the derivative and solve for zero to find the local max and mins.
f'(x) = 12x^2 + 36x - 216
0 = 12(x^2 + 3x - 18)
0 = 12(x + 6)(x - 3)
x = -6 OR x = 3
Given the shape of a positive quartic function, we know that the first would be a maximum and the second would be a minimum.
As for the increasing, we know that a third power, positive function starts down and increases to the local maximum. It also increases after the local min. The rest of the time it would be decreasing.
In order to find the inflection point, we take a derivative of the derivative and then solve for zero.
f'(x) = 12(x^2 + 3x - 18)
f''(x) = 2x + 3
0 = 2x + 3
-3 = 2x
-3/2 = x
