Example Question - local maximum
Here are examples of questions we've helped users solve.
Local Maximum of a Function
题目要求我们找出函数 \( f \) 在哪一个区间上保证有一个局部最大值。 首先，根据表中给出的 \( f' \) 的值，我们可以分析函数 \( f \) 的增减性。函数在其导数为正的区间上增加，在导数为负的区间上减少。因此，一个局部最大值出现在导数从正变成负的点上。 根据表格，我们可以看出： - 在 \( x = 1 \) 时前后导数的符号从正（\( f'(0) = 1 \)）变为负（\( f'(1) = -2 \)），因此 \( x = 1 \) 可能是局部最大值点。 - \( x = 2 \) 到 \( x = 3 \) 时，导数一直为负，因此在 \( (2,3) \) 区间内不可能有局部最大值。 - \( x = 3 \) 到 \( x = 4 \) 时，导数符号仍然为负。 - 在 \( x = 4 \) 时，导数符号并没有改变。虽然 \( f'(4) = 0 \)，但导数的符号没有变化，所以 \( x = 4 \) 不是局部最大值点。 所以，函数 \( f \) 保证在 \( x = 1 \) 处有局部最大值，对应的区间是 \( (0,1) \)。选择答案是 (A)。 这种问题的关键在于分析导数的符号变化，来找到函数从增加转变为减少的点，这些点可能是局部最大值点。
Identifying Local Maximum with First Derivative
根据题目中给出的一阶导数 \(f'(x)\) 的值，我们可以判断函数 \(f(x)\) 在哪个区间上可能达到局部最大值。 局部最大值出现在一阶导数由正变负的点。根据表格，当 \(x\) 从 0 变到 1 时，\(f'(x)\) 由 1 变为 -2，所以这是由正到负的变化。因此，在 \(x = 1\) 处 \(f(x)\) 可能存在局部最大值。所以我们关心的区间至少包括这一点。 因此，答案是选项 (B) (1, 2)，因为这个区间包含了 \(x = 1\) 这点，在 \(x = 1\) 处函数 \(f(x)\) 从增加变为减少，所以这里可能有一个局部最大值。其他区间，包括 (0, 1)、(2, 4) 和 (3, 5)，要么不包含这一点，要么 \(f'(x)\) 的符号没有从正变为负。所以这些区间都不是我们寻找的区间。
Determining Interval for Local Maximum of a Function
The question asks us to find on which interval the function f is guaranteed to attain a local maximum, based on the selected values of its first derivative f'(x) given in the table. A local maximum occurs where the derivative changes from positive to negative. Looking at the table of values for f'(x), we see: - f'(0) = -1, meaning the derivative is negative just before x=0. - f'(1) = 2, meaning the derivative is positive at x=1. - f'(2) = -3, meaning the derivative is negative at x=2. - f'(3) = 0, meaning the derivative is zero at x=3. - f'(4) = 5, meaning the derivative is positive at x=4. From these values, we notice that f'(x) goes from positive at x=1 to negative at x=2. This indicates that there is a local maximum somewhere between x=1 and x=2, as per the first derivative test which says that if the derivative changes from positive to negative at some point, then this point is a local maximum. Therefore, the correct interval on which f is guaranteed to attain a local maximum is: (B) (1,2)
Function Analysis from Hand-Drawn Graph
The image shows a hand-drawn graph of a function f(x) and also contains two handwritten questions in Turkish. Before I proceed to translate and answer your questions, I'll describe the graph of the function f(x) as per the provided image: The graph appears to depict a continuous function f(x) with the following characteristics: - The function has a local maximum at x = 2. - The function has a local minimum at some x-value between 3 and 4. - The function has x-intercepts at x = 1, 2, and 5. - There is a point of inflection where the concavity of the function changes around x = 3. The handwritten questions in Turkish ask for the following: a) Determine the intervals in which the function is increasing and decreasing (Artan ve azalan aralıkları belirleyin). b) Find the maximum and minimum points of the function (Maksimum ve minimum noktalarını bulun). Let me address each part based on the provided graph: a) Increasing and Decreasing Intervals: - The function is increasing from the left of the graph up to x = 2, where it reaches a local maximum. - Then the function is decreasing from x = 2 to approximately x = 3.5 (assuming that the local minimum appears around there, as the exact x-value is unclear from the sketch). - After this local minimum, the function starts increasing again until it goes off the right edge of the graph. So, the intervals in which the function is increasing and decreasing are: - Increasing: (-∞, 2) and (approximately 3.5, +∞) - Decreasing: (2, approximately 3.5) b) Maximum and Minimum Points: - The local maximum is at x = 2. The exact value of f(2) is not provided, but based on the graph, it is below 5/2 as indicated on the graph. - The local minimum occurs between x = 3 and x = 4. The exact coordinates of the minimum are unable to be determined without more information or a clearer graph. Please note that the answers are based on the interpretation of the drawn graph, which may not be perfectly scaled or accurate.
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