Home Practice Questions AP Calculus AB Analytical Applications of Differentiation Question No. 19

Answer is: option4

Solution:

We are given the function:

\[ y = ax^3 - 6x^2 + bx - 4 \]

First Derivative:

\[ y' = 3ax^2 - 12x + b \]

Second Derivative:

\[ y'' = 6ax - 12 \]

A point of inflection occurs where \( y'' = 0 \). We are given that \( (2,-2) \) is a point of inflection, so set \( x = 2 \):

\[ 6a(2) - 12 = 0 \]

\[ 12a - 12 = 0 \]

\[ 12a = 12 \Rightarrow a = 1 \]

Substituting into the original equation:

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

\[ -2 = (1)(2^3) - 6(2^2) + b(2) - 4 \]

\[ -2 = 8 - 24 + 2b - 4 \]

\[ -2 = -20 + 2b \]

\[ 2b = 18 \Rightarrow b = 9 \]

Thus,

\[ a + b = 1 + 9 = 10 \]

Final Answer: 10