`f(x) = x^3 - 6x^2 + 12x` Find the points of inflection and discuss the concavity of the graph of the function.

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Chapter 3, 3.4 - Problem 15 - Calculus of a Single Variable (10th Edition, Ron Larson).
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mathace | (Level 3) Assistant Educator

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Given: `f(x)=x^3-6x^2+12x`

Find the critical values for x by setting the second derivative of the function equal to zero and solving for the x value(s).





The critical value for the second derivative is x=2.

If f''(x)>0, the curve is concave up in the interval.

If f''(x)<0, the curve is concave down in the interval.

Choose a value for x that is less than 2.

f''(0)=-12 Since f''(0)<0 the graph is concave down in the interval (-`oo, 2).`

Choose a value for x that is greater that 2.

f''(3)=6 Since f''(3)>0 the graph is concave up in the interval (2, `oo).`

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