Let h be a twice differentiable function, and let h(8)=5, h^(')(8)=0, and h^('')(8)=-4. What occurs in the graph of h at the point (8,5) ? Choose 1 answer: (A) (8,5) is a minimum point. (B) (8,5) is a maximum point. (C) There's not enough information to tell.

Question

Let  h be a twice differentiable function, and let  h(8)=5,  h^(')(8)=0, and  h^('')(8)=-4. What occurs in the graph of  h at the point  (8,5) ? Choose 1 answer: (A)  (8,5) is a minimum point. (B)  (8,5) is a maximum point. (C) There's not enough information to tell.

Bytelearn · Accepted Answer

Given Information: To determine what occurs at the point (8,5) on the graph of h, we need to analyze the given information about the function h and its derivatives at the point x=8. Given: - h(8) = 5, which means the function passes through the point (8,5). - h'(8) = 0, which indicates that the slope of the tangent line to the graph of h at x=8 is zero. This suggests that (8,5) could be a local maximum, a local minimum, or a point of inflection. - h''(8) = -4, which tells us the concavity of the function at x=8. Since h''(8) is negative, the graph of h is concave down at x=8.Analysis: Using the second derivative test, we can determine whether (8,5) is a local maximum or minimum. If the second derivative at a point where the first derivative is zero is negative, the function has a local maximum at that point. If the second derivative is positive, the function has a local minimum at that point. Given that h''(8) = -4, which is negative, we conclude that (8,5) is a local maximum point.

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