To find which cubic function has a point of inflection at \((-4, 2)\), we need to understand what a point of inflection means. A point of inflection occurs where the second derivative of the function changes sign, which typically implies that the second derivative is zero at that point. Additionally, the original function must pass through the point given.
Let's analyze the given functions. We will first verify that the candidate functions pass through the point \((-4, 2)\), and then we will differentiate them to check for a point of inflection.
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\( h(x) = (x - 4)^3 + 2 \)
- \( h(-4) = (-4 - 4)^3 + 2 = (-8)^3 + 2 = -512 + 2 = -510 \) (does not pass through (−4, 2)).
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\( h(x) = -2(x + 4)^3 + 2 \)
- \( h(-4) = -2(-4 + 4)^3 + 2 = -2(0) + 2 = 2 \) (passes through (−4, 2)).
- Now let's find the second derivative:
- First derivative: \( h'(x) = -6(x + 4)^2 \)
- Second derivative: \( h''(x) = -12(x + 4) \) and \( h''(-4) = -12(0) = 0 \) (point of inflection).
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\( h(x) = 2(x + 4)^3 - 2 \)
- \( h(-4) = 2(-4 + 4)^3 - 2 = 2(0) - 2 = -2 \) (does not pass through (−4, 2)).
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\( h(x) = 3(x - 4)^3 - 2 \)
- \( h(-4) = 3(-4 - 4)^3 - 2 = 3(-8)^3 - 2 = 3(-512) - 2 = -1536 - 2 = -1538 \) (does not pass through (−4, 2)).
The function that passes through the point \((-4, 2)\) and has a point of inflection at that location is:
\( h(x) = -2(x + 4)^3 + 2 \).
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