AP Calculus AB / Analytical Applications of Differentiation / Question No. 19

19. If the graph of $y = a x^{3} - 6 x^{2} + b x - 4$ has a point of inflection at $(2, - 2)$ , what is the value of $a + b$ ?

-2

Answer is: option4

Solution:

We are given the function:

$y = a x^{3} - 6 x^{2} + b x - 4$

First Derivative:

$y^{'} = 3 a x^{2} - 12 x + b$

Second Derivative:

$y^{″} = 6 a x - 12$

A point of inflection occurs where $y^{″} = 0$ . We are given that $(2, - 2)$ is a point of inflection, so set $x = 2$ :

$6 a (2) - 12 = 0$

$12 a - 12 = 0$

$12 a = 12 \Rightarrow a = 1$

Substituting into the original equation:

Since the point $(2, - 2)$ lies on the curve, substitute $x = 2$ and $y = - 2$ :

$- 2 = (1) (2^{3}) - 6 (2^{2}) + b (2) - 4$

$- 2 = 8 - 24 + 2 b - 4$

$- 2 = - 20 + 2 b$

$2 b = 18 \Rightarrow b = 9$

Thus,

$a + b = 1 + 9 = 10$

Final Answer: 10

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