What are the points of inflection of f(x)= x^3 - 12x^2 + 2x + 15x ?

Nov 16, 2016

$\text{(4 , -30)}$ is the inflection point.

Explanation:

Determining the points of inflection is by finding the second
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derivative then solve the equation for $f ' ' \left(x\right) = 0$.
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$f ' \left(x\right) = 3 {x}^{2} - 24 x + 2 + 15$
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$f ' ' \left(x\right) = 6 x - 24$
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$f ' ' \left(x\right) = 0$
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$6 x - 24 = 0$
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$\Rightarrow 6 x = 24$
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$\Rightarrow x = \frac{24}{6} = 4$
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Finding its ordinate $f \left(4\right)$.
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$f \left(4\right) = {4}^{3} - 12 {\left(4\right)}^{2} + 2 \left(4\right) + 15 \left(4\right)$
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$f \left(4\right) = 64 - 192 + 8 + 90$
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$f \left(4\right) = - 30$
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Hence, $\text{(4 , -30)}$ is the inflection point.