
Thomas R. answered 06/03/18
Tutor
4.9
(1,014)
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