Find the slope of the tangent at the point of inflection of $y = x^{3} - 3 x^{2} + 6 x + 2000.$

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Solution

We have,

$y = x^{3} - 3 x^{2} + 6 x + 2000$

On differentiation and we get,

$y^{'} = 3 x^{2} - 6 x + 6$

Again, differentiating and we get,

$y^{''} = 6 x - 6$

For points of inflection

$y^{''} = 0$

$6 x - 6 = 0$

$6 x = 6$

$x = 1$

Put the value of x in given function and we get,

$y = 1^{3} - 3 \times 1^{2} + 6 \times 1 + 2000$

$y = 2004$
Hence, the point of inflection is $(1, 2004)$ .

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