How do you find the inflection points of the graph of the function f(x) = x^3 - 3x^2 + 3x?

2 Answers
Apr 6, 2015

Find the points on the graph where the concavity changes.

f(x) = x^3 - 3x^2 + 3x.

So, f'(x) = 3x^2 - 6x + 3.

And f''(x) = 6x - 6.

f''(x)=0 for x=1. Testing on each side of 1 we find that

f''(x) < 0 (so the graph of f is concave down) for x<1
f''(x) > 0 (so the graph of f is concave up) for x>1

At x=1, we have y=f(1)=3-3+1=1.

The inflection point is (1, 1).

Apr 6, 2015

You must study your second derivative:
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