Let h be a twice differentiable function, and let h(5)=1, h^(')(5)=0, and h^('')(5)=2. What occurs in the graph of h at the point (5,1) ? Choose 1 answer: (A) (5,1) is a minimum point. (B) (5,1) is a maximum point. (C) There's not enough information to tell.

Question

Let  h be a twice differentiable function, and let  h(5)=1,  h^(')(5)=0, and  h^('')(5)=2. What occurs in the graph of  h at the point  (5,1) ? Choose 1 answer: (A)  (5,1) is a minimum point. (B)  (5,1) 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 (5,1) on the graph of h, we need to analyze the given information about the function h and its derivatives at x = 5. Given: h(5) = 1, which means the point (5,1) lies on the graph of h. h'(5) = 0, which indicates that the slope of the tangent to the graph of h at x = 5 is zero. This means that the graph has a horizontal tangent line at x = 5. h''(5) = 2, which tells us that the concavity of the graph of h at x = 5 is upwards since the second derivative is positive.Interpret First Derivative: Using the information from the first derivative, h'(5) = 0, we know that the point (5,1) could be a local maximum, a local minimum, or a point of inflection. However, we cannot conclude which one it is based solely on the first derivative.Determine Concavity: The second derivative, h''(5) = 2, is positive, which means the graph of h is concave up at x = 5. In the context of the first derivative being zero, this indicates that the point (5,1) is a local minimum because the graph is shaped like a ＂U＂ at that point.

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