SOLUTION: find the point of inflection f(x)= 1/4x^4 - x^3 + 9

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Question 288738: find the point of inflection
f(x)= 1/4x^4 - x^3 + 9
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
f(x)= %281%2F4%29x%5E4+-+x%5E3+%2B+9
To find points of inflection, we need the second derivative:
f'(x)= 4%2A%281%2F4%29x%5E%284-1%29+-+3%2Ax%5E%283-1%29+=+x%5E3+-+3x%5E2
f''(x)= 3%2Ax%5E%283-1%29+-+2%2A3x%5E%282-1%29+=+3x%5E2+-6x

Now we need to find the value(s) of x that make the second derivative zero:
0+=+3x%5E2+-6x
0+=+3x%28x+-+2%29
3x = 0 or x-2 = 0
x = 0 or x = 2

These x values are only candidates for points of inflection. Points of inflection are where the curvature of the graph changes from positive (i.e. concave up) to negative (concave down) or vice versa. So we'll check to see if there is such a change for each of these candidates:

For x = 0 we will try x values on either side. We''l try -1 and 1
f''(-1) = 3(-1)((-1)-2) = (-3)(-9) = 27
f''(1) = 3(1)((1)-2) = (3)(-1) = -3
As we can see, the concavity does change so x=0 is a point of inflection. To find the y coordinate of this point of inflection we find f(0). You should find that f(0) = 9 so a point of inflection is (0, 9).

For x = 2 we will try x values on either side. We'll try 1 and 3. Earlier we already found f''(1) = -3.
f''(3) = 3(3)((3)-2) = (9)(1) = 9
So the concavity changes on either side of x=2, also. So we have a second point of inflection. For its y coordinate we find f(2). You should find that f(2) = 5. So the second point of inflection is (2, 5).

Here's a graph to illustrate:
graph%28400%2C+400%2C+-2%2C+12%2C+-2%2C+12%2C+%281%2F4%29x%5E4+-+x%5E3+%2B+9%29