# `f(x) = (x - 2)^3(x - 1)` Find the points of inflection and discuss the concavity of the graph of the function.

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Expert Answers

Borys Shumyatskiy | Certified Educator

f is infinite differentiable and its concavity is fully described by the sign of f''.

`f'(x) = 3(x-2)^2*(x-1) + (x-2)^3 =`

`(x-2)^2*(3x-3+x-2) = (x-2)^2*(4x-5).`

`f''(x) = 4*(x-2)^2 + 2(x-2)*(4x-5) =`

`2*(x-2)*(2x-4+4x-5) = 2*(x-2)*(6x-9).`

This function is negative on (1.5, 2) and positive on `(-oo, 1.5)` and on `(2, +oo).` Therefore f is concave upward on `(-oo, 1.5) uu (2, +oo)` and concave downward on (1.5, 2). The points of inflection are **1.5** and **2**.