What are the points of inflection of f(x)= 10(x-5)^3+2?

Feb 4, 2016

$\left(5 , 2\right)$

Explanation:

Points of inflection, where concavity changes, occur when the second derivative of the function is zero, that is, when $f ' ' \left(x\right) = 0$.

We may find the second derivative by application of the power rule :

$\therefore f ' \left(x\right) = 30 {\left(x - 5\right)}^{2}$

$\therefore f ' ' \left(x\right) = 60 \left(x - 5\right)$.

$\therefore f ' ' \left(x\right) = 0 \iff x = 5$.

But $f \left(5\right) = 10 {\left(5 - 5\right)}^{3} + 2 = 2$.

Hence the inflection point is $\left(5 , 2\right)$.