Let f(x)= x^3-9x^2 +9x-5. Find the open intervals on which f is concave up (down). Then determine the x-coordi
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HOME > > Let f(x)= x^3-9x^2 +9x-5. Find the open intervals on which f is concave up (down). Then determine the x-coordi

Let f(x)= x^3-9x^2 +9x-5. Find the open intervals on which f is concave up (down). Then determine the x-coordi

[From: ] [author: ] [Date: 12-05-29] [Hit: ]
3.When we see the word concavity, we know to look for the second derivative.We need to find where the second derivative changes sign, because this will tell us where our points of inflection are.As such,......
let f(x)= x^3-9x^2 +9x-5. Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inflection points of f.

1.  
f is concave up on the intervals

2.  
f is concave down on the intervals

3.  
The inflection points occur at x =

-
f(x) = x^3 - 9x^2 + 9x - 5

When we see the word concavity, we know to look for the second derivative.

f'(x) = 3x^2 - 18x + 9

f''(x) = 6x - 18

We need to find where the second derivative changes sign, because this will tell us where our points of inflection are.

As such, we set f''(x) = 0

f''(x) = 0
6x-18 = 0
6x = 18
x = 3

Now, we test to the left and right of x = 3 to see where f''(x) is positive and where it is negative, as it is 0 at x = 3.

Picking random points we find:

f''(0) = 6(0) - 18 = -18
f'(5) = 6(5) - 18 = 30 - 18 = 12

Clearly, f''(x) is negative to the left or x = 3, and positive to the right of x = 3. We just simply need to know the f''(x) > 0 implies upward concavity while f''(x) < 0 implies downward concavity.

Answers:

1) (3, ∞)

2) (-∞, 3)

3) x = 3
1
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