Question

Which of the following cubic functions has a point of inflection of (2,−5)
?(1 point)
Responses

g(x)=−3(x+2)3−5
g left parenthesis x right parenthesis equals negative 3 left parenthesis x plus 2 right parenthesis cubed minus 5

g(x)=2(x−2)3−5
g left parenthesis x right parenthesis equals 2 left parenthesis x minus 2 right parenthesis cubed minus 5

g(x)=(x+2)3+5
g left parenthesis x right parenthesis equals left parenthesis x plus 2 right parenthesis cubed plus 5

g(x)=−2(x−2)3+5

Answer 1

To find which cubic function has a point of inflection at (2, -5), we first need to remember that a point of inflection occurs where the second derivative changes sign, which typically coincides with the roots of the second derivative of the function.

Let's analyze each function and check if they have an inflection point at \( x = 2 \) and whether they pass through the point \( (2, -5) \).

1. **g(x) = -3(x + 2)³ - 5**
- Evaluate at \( x = 2 \):
\[
g(2) = -3(2 + 2)^3 - 5 = -3(4)^3 - 5 = -3(64) - 5 = -192 - 5 = -197
\]
- Not equal to -5.

2. **g(x) = 2(x - 2)³ - 5**
- Evaluate at \( x = 2 \):
\[
g(2) = 2(2 - 2)^3 - 5 = 2(0)^3 - 5 = 0 - 5 = -5
\]
- Equal to -5. Now let's find the second derivative to check for the inflection point.
- First derivative:
\[
g'(x) = 2 \cdot 3(x - 2)^2 = 6(x - 2)^2
\]
- Second derivative:
\[
g''(x) = 6 \cdot 2(x - 2) = 12(x - 2)
\]
- Set the second derivative to zero:
\[
12(x - 2) = 0 \Rightarrow x = 2
\]
- The second derivative changes sign at \( x = 2 \). So, (2, -5) is a point of inflection.

3. **g(x) = (x + 2)³ + 5**
- Evaluate at \( x = 2 \):
\[
g(2) = (2 + 2)^3 + 5 = (4)^3 + 5 = 64 + 5 = 69
\]
- Not equal to -5.

4. **g(x) = -2(x - 2)³ + 5**
- Evaluate at \( x = 2 \):
\[
g(2) = -2(2 - 2)^3 + 5 = -2(0)^3 + 5 = 0 + 5 = 5
\]
- Not equal to -5.

The only function that has a point of inflection at \( (2, -5) \) is:

**g(x) = 2(x - 2)³ - 5**.