How do you find the inflection points of the graph of the function: y=x^3-15x^2+33x+100?

Jul 22, 2015

The inflection point is $\left(x , y\right) = \left(5 , 15\right)$.

Explanation:

If $f \left(x\right) = {x}^{3} - 15 {x}^{2} + 33 x + 100$, then the first derivative is $f ' \left(x\right) = 3 {x}^{2} - 30 x + 33$ and the second derivative is $f ' ' \left(x\right) = 6 x - 30 = 6 \left(x - 5\right)$. The second derivative changes sign (from negative to positive) as $x$ increases through $x = 5$, making this the first coordinate of the unique inflection point. In this case, the inflection point occurs where the graph of $y = f \left(x\right)$ changes from concave down (frown-like) to concave up (smile-like).

The second coordinate of the inflection point is $f \left(5\right) = {5}^{3} - 15 \cdot {5}^{2} + 33 \cdot 5 + 100$

$= 125 - 375 + 165 + 100 = 15$.

The inflection point is therefore $\left(x , y\right) = \left(5 , f \left(5\right)\right) = \left(5 , 15\right)$