Let f be a twice differentiable function, and let f(-7)=6, f^(')(-7)=0, and f^('')(-7)=-5. What occurs in the graph of f at the point (-7,6) ? Choose 1 answer: (A) (-7,6) is a minimum point. (B) (-7,6) is a maximum point. (C) There's not enough information to tell.

Question

Let  f be a twice differentiable function, and let  f(-7)=6,  f^(')(-7)=0, and  f^('')(-7)=-5. What occurs in the graph of  f at the point  (-7,6) ? Choose 1 answer: (A)  (-7,6) is a minimum point. (B)  (-7,6) is a maximum point. (C) There's not enough information to tell.

Bytelearn · Accepted Answer

Given Information Analysis: To determine what occurs at the point (-7,6) on the graph of f, we need to analyze the given information about the function and its derivatives at x = -7. Given: f(-7) = 6, f'(-7) = 0, and f''(-7) = -5. The value f(-7) = 6 tells us that the point (-7,6) lies on the graph of f. The derivative f'(-7) = 0 indicates that the slope of the tangent to the graph of f at x = -7 is zero, which means the graph has a horizontal tangent line at this point. The second derivative f''(-7) = -5 tells us about the concavity of the graph at x = -7. Since f''(-7) is negative, the graph of f is concave down at this point.Point Characteristics: Now, we can determine what type of point (-7,6) is on the graph of f. A point where the first derivative is zero and the second derivative is negative is typically a local maximum. This is because the horizontal tangent line indicates a potential extremum, and the concave down nature (negative second derivative) suggests that the graph is curving downwards, making it a peak or a maximum point.

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