Let h be a twice differentiable function, and let h(-4)=-3, h^(')(-4)=0, and h^('')(-4)=0. What occurs in the graph of h at the point (-4,-3) ? Choose 1 answer: (A) (-4,-3) is a minimum point. (B) (-4,-3) is a maximum point. (C) There's not enough information to tell.

Question

Let  h be a twice differentiable function, and let  h(-4)=-3,  h^(')(-4)=0, and  h^('')(-4)=0. What occurs in the graph of  h at the point  (-4,-3) ? Choose 1 answer: (A)  (-4,-3) is a minimum point. (B)  (-4,-3) is a maximum point. (C) There's not enough information to tell.

Bytelearn · Accepted Answer

Given Information: We are given that h is a twice differentiable function, and we have information about the function and its derivatives at the point x = -4. The value of the function at x = -4 is h(-4) = -3, which tells us that the point (-4, -3) lies on the graph of h. First Derivative Analysis: The first derivative of h at x = -4 is given as h'(-4) = 0. The first derivative represents the slope of the tangent line to the graph of the function at a particular point. Since h'(-4) = 0, the tangent line to the graph of h at x = -4 is horizontal. This could indicate a local maximum, a local minimum, or a point of inflection. Second Derivative Analysis: The second derivative of h at x = -4 is given as h''(-4) = 0. The second derivative provides information about the concavity of the function. If the second derivative is positive, the function is concave up (shaped like a cup), and if it is negative, the function is concave down (shaped like a cap). Since h''(-4) = 0, we cannot determine the concavity of the function at x = -4 from this information alone. Conclusion: Given that both the first and second derivatives of h at x = -4 are zero, we cannot conclusively determine whether (-4, -3) is a minimum point, a maximum point, or neither. We would need additional information about the behavior of the function and its derivatives in the vicinity of x = -4 to make a determination.

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