Let g be a twice differentiable function, and let g(-5)=-6, g^(')(-5)=0, and g^('')(-5)=0. What occurs in the graph of g at the point (-5,-6) ? Choose 1 answer: (A) (-5,-6) is a minimum point. (B) (-5,-6) is a maximum point. (C) There's not enough information to tell.

Question

Let  g be a twice differentiable function, and let  g(-5)=-6,  g^(')(-5)=0, and  g^('')(-5)=0. What occurs in the graph of  g at the point  (-5,-6) ? Choose 1 answer: (A)  (-5,-6) is a minimum point. (B)  (-5,-6) is a maximum point. (C) There's not enough information to tell.

Bytelearn · Accepted Answer

Given Information: We are given that g is a twice differentiable function, and we have the following information about the function at x = -5: g(-5) = -6, g'(-5) = 0, and g''(-5) = 0. The value g(-5) = -6 tells us the point (-5, -6) lies on the graph of g.Tangent Slope at x = -5: The derivative g'(-5) = 0 indicates that the slope of the tangent to the graph of g at x = -5 is zero. This means that the graph has a horizontal tangent line at x = -5.Concavity Information: The second derivative g''(-5) = 0 provides information about the concavity of the graph at x = -5. However, since g''(-5) = 0, we cannot determine whether the graph is concave up or concave down at this point. To classify the point as a minimum or maximum, we would need g''(-5) to be positive (indicating concave up and thus a minimum) or negative (indicating concave down and thus a maximum).Conclusion: Since we do not have information about the concavity being strictly positive or negative, we cannot conclude whether (-5, -6) is a minimum or maximum point. We need additional information about the behavior of g'' around x = -5 to make a definitive conclusion.

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