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

Question

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

Bytelearn · Accepted Answer

Analyze Function Information: To determine what occurs at the point (-1,4) on the graph of the function g, we need to analyze the given information about the function's value and its derivatives at the point x = -1.Point (-1,4) on Graph: The function g is given to have a value of g(-1) = 4. This tells us that the point (-1,4) lies on the graph of g, but it does not tell us anything about the nature of this point (whether it's a maximum, minimum, or neither).First Derivative Analysis: The first derivative of g at x = -1 is given as g'(-1) = 0. This indicates that the slope of the tangent to the graph of g at x = -1 is zero, which means the graph has a horizontal tangent line at this point. This could be indicative of a local maximum, local minimum, or a point of inflection.Second Derivative Analysis: The second derivative of g at x = -1 is given as g''(-1) = 3. Since the second derivative is positive, it tells us that the graph of g is concave up at x = -1. This concavity, combined with the horizontal tangent line, indicates that the point (-1,4) is a local minimum.

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