The graph of f"(x), the second derivative of f, is shown below. If the point (4, f(4)) is a critical number for f(x), is it a local min, local max or neither? y =f"(x) 6 -5 -4 -3 -2 -1 1 2 3 4/5 6 7 a) O Neither b) O Local minimum c) Local maximum
The graph of f"(x), the second derivative of f, is shown below. If the point (4, f(4)) is a critical number for f(x), is it a local min, local max or neither? y =f"(x) 6 -5 -4 -3 -2 -1 1 2 3 4/5 6 7 a) O Neither b) O Local minimum c) Local maximum
The graph of f"(x), the second derivative of f, is shown below. If the point (4, f(4)) is a critical number for f(x), is it a local min, local max or neither? y =f"(x) 6 -5 -4 -3 -2 -1 1 2 3 4/5 6 7 a) O Neither b) O Local minimum c) Local maximum
Hi. If by second derivative test, if x = a is a critical point of f(x) and f''(a) > 0, then (a, f(a)) is a point of local minimum. Can you explain why my answer is wrong?
Thank you so much!
Transcribed Image Text:**Description of the Image:**
The graph presents \( f''(x) \), the second derivative of a function \( f \). The equation of the curve is \( y = f''(x) \).
**Axes:**
- The horizontal axis (x-axis) ranges from approximately -6 to 8.
- The vertical axis (y-axis) is not labeled with numbers, but it crosses the x-axis around zero for guidance.
**Curve Details:**
- The graph indicates an oscillating curve with a general shape similar to a cubic function.
- The curve descends to a minimum around \( x = -3.5 \) and then ascends, crossing the x-axis at \( x = 0 \).
- The curve peaks before crossing the x-axis again at \( x = 4 \) and descends, crossing once more shortly after.
**Critical Point Analysis:**
- Given the point (4, \( f(4) \)) as a critical number for \( f(x) \):
- Since \( f''(4) > 0 \), the point is a local minimum for \( f(x) \).
**Options:**
- a) Neither (not selected)
- b) **Local minimum** (selected)
- c) Local maximum (not selected)
Formula Formula A function f ( x ) is also said to have attained a local minimum at x = a , if there exists a neighborhood ( a − δ , a + δ ) of a such that, f ( x ) > f ( a ) , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a f ( x ) − f ( a ) > 0 , ∀ x ∈ ( a − δ , a + δ ) , x ≠ a In such a case f ( a ) is called the local minimum value of f ( x ) at x = a .
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