Let h be a twice differentiable function, and let h(6)=7, h^(')(6)=0, and h^('')(6)=0. What occurs in the graph of h at the point (6,7) ? Choose 1 answer: (A) (6,7) is a minimum point. (B) (6,7) is a maximum point. (C) There's not enough information to tell.

Question

Let  h be a twice differentiable function, and let  h(6)=7,  h^(')(6)=0, and  h^('')(6)=0. What occurs in the graph of  h at the point  (6,7) ? Choose 1 answer: (A)  (6,7) is a minimum point. (B)  (6,7) is a maximum point. (C) There's not enough information to tell.

Bytelearn · Accepted Answer

Analyze Given Information: To determine what occurs at the point (6,7) on the graph of h, we need to analyze the given information about the function and its derivatives at x = 6. Given: h(6) = 7, h'(6) = 0, and h''(6) = 0. The value h(6) = 7 simply gives us the point on the graph, which is (6,7). The first derivative h'(6) = 0 indicates that the slope of the tangent line to the graph of h at x = 6 is zero. This means that the graph has a horizontal tangent at this point, which could be a minimum, maximum, or a point of inflection. The second derivative h''(6) = 0 provides information about the concavity of the graph at x = 6. Since the second derivative is zero, we cannot determine whether the graph is concave up or concave down at this point.Interpret First Derivative: To determine whether (6,7) is a minimum or maximum point, or if there's not enough information, we would typically look at the sign of the second derivative. If h''(6) > 0, the graph would be concave up, and (6,7) would be a minimum point. If h''(6) < 0, the graph would be concave down, and (6,7) would be a maximum point. However, since h''(6) = 0, we cannot conclude whether the graph is concave up or down at that point. Therefore, we do not have enough information to determine if (6,7) is a minimum or maximum point.

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