Thomas R. answered • 06/03/18

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A.S. Mathematics, experienced and creative

Michelle:

f(x) = X³ +3X² - 144X + 1

To find the behaviors you need, you must first derive it. That's simple. Power rule all the way. That gets you:

f'(X) = 3X² + 6X - 144

Set it equal to zero and solve for X:

3X² + 6X - 144 = 0

3(X² + 2X - 48) = 0

3(X + 8)(X - 6) = 0

X = -8 , X = 6

These are the critical values. The function turns horizontal at each of these. To find out what it does before or after each one, we need the second derivative:

f''(X) = 6X + 6

f''(-8) = 6(-8) + 6 = -48 + 6 = -42

f''(6) = 6(6) + 6 = 36 + 6 = 42

This means that at X = -8, it's concave down and at X = 6 it's concave up. How do we know when it changes from one to the other? Inflection point(s). Set the f'' to zero:

6X + 6 = 0

6X = -6

X = -1

Now we know it was concave down from -∞ to -1, and concave up from -1 to +∞. That in turn means it is:

rising from -∞ to -8

falling from -8 to 6

rising from 6 to +

infinity