Answer:
increasing from (-∞, -3)∪(3, ∞)
decreasing from (-3, 3)
concave up from (0, ∞)
concave down from (-∞, 0)
Step-by-step explanation:
The derivative gives you the slope of the function, so to find where the graph is increasing, find the derivative of f(x).
- A positive derivative, f'(x)>0, means it's increasing,
- A zero derivative, f'(x)=0, means the slope is flat.
- A negative derivative, f'(x)<0, means it's decreasing
So solve for when the derivative is 0 and plug in random number into x and see if it's positive or not.
I had x=-3 and x=3 for f'(x)=0. So i plug a number below x=-3
I chose x=-∞ and got f'(-∞)=(-∞)²-27 = ∞ which is a positive number, so the slope is increase when x<3.
Then I chose a number between x=3 and x=3.
I chose x=0 and got f'(0)=(0)²-27 =-27 which is a negative number, so the slope is decreasing when -3<x<3.
Then I chose a number greater than x=3.
I chose x=∞ and got f'(∞)=(∞)²-27 = ∞ which is a positive number, so the slope is increase when x>3.
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To find if it concave up or down, find the second derivative. The solving process is the same as described in the previous paragraphs.
- If the second derivative is positive, f''(x)>0, it concave up
- If the second derivative is zero, f''(x)=0, it doesn't concave at all
- If the second derivative is negative, f''(x)<0, it concave down
