If a function \( f \) is continuous for all \( x \) and if \( f \) has a relative maximum at \((-1, 4)\) and a relative minimum at \((3, -2)\), which of the following statements must be true?
A. The graph of \( f \) has a point of inflection somewhere between \( x = -1 \) and \( x = 3 \).
B. \( f'(-1) = 0 \).
C. This is wrong.
D. The graph of \( f \) has a horizontal tangent line at \( x = 3 \).
E. The graph of \( f \) intersects both axes.
I understand why E is correct, but I do not get why A, B, and D are all wrong. Aren't B and D to be expected since they are relative max/min's? I also can't imagine a case in which "A" is incorrect. Can someone explain why they are wrong? Thank you!
Answer (2)
The statements a, b, and d are not necessarily true in this case. However, option e is true. The graph of f must intersect both axes because the **function **is continuous. ;
The statements A, B, and D are not necessarily true regarding the function's behavior at its maximum and minimum points. Option E is true, as the continuous function must intersect both axes. Therefore, the selected answer is C: This is wrong.
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