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Let 
g be a twice differentiable function, and let 
g(-6)=-1, 
g^(')(-6)=0, and 
g^('')(-6)=-3.
What occurs in the graph of 
g at the point 
(-6,-1) ?
Choose 1 answer:
(A) 
(-6,-1) is a minimum point.
(B) 
(-6,-1) is a maximum point.
(C) There's not enough information to tell.

Let g g be a twice differentiable function, and let g(6)=1 g(-6)=-1 , g(6)=0 g^{\prime}(-6)=0 , and g(6)=3 g^{\prime \prime}(-6)=-3 .\newlineWhat occurs in the graph of g g at the point (6,1) (-6,-1) ?\newlineChoose 11 answer:\newline(A) (6,1) (-6,-1) is a minimum point.\newline(B) (6,1) (-6,-1) is a maximum point.\newline(C) There's not enough information to tell.

Full solution

Q. Let g g be a twice differentiable function, and let g(6)=1 g(-6)=-1 , g(6)=0 g^{\prime}(-6)=0 , and g(6)=3 g^{\prime \prime}(-6)=-3 .\newlineWhat occurs in the graph of g g at the point (6,1) (-6,-1) ?\newlineChoose 11 answer:\newline(A) (6,1) (-6,-1) is a minimum point.\newline(B) (6,1) (-6,-1) is a maximum point.\newline(C) There's not enough information to tell.
  1. Analyze Given Information: First, we analyze the given information about the function gg at the point x=6x = -6. We are given that g(6)=1g(-6) = -1, which means the point (6,1)(-6, -1) lies on the graph of gg.
  2. Consider First Derivative: Next, we consider the first derivative g(6)=0g'(-6) = 0. This indicates that the slope of the tangent to the graph of gg at x=6x = -6 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 saddle point (inflection point).
  3. Look at Second Derivative: We then look at the second derivative g(6)=3g''(-6) = -3. Since the second derivative is negative, it tells us that the graph of gg is concave down at x=6x = -6. This concavity, combined with the horizontal tangent line, means that the point (6,1)(-6, -1) is a local maximum.
  4. Conclude Maximum Point: Given the information about the first and second derivatives at x=6x = -6, we can conclude that (6,1)(-6, -1) is a maximum point on the graph of gg.

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