Sketch the graph and show all local extrema and inflection points.

y = x4

Question

Sketch the graph and show all local extrema and inflection points.

y = x4

BlackTom AI · Accepted Answer

To evaluate the question, we first recall the basic shape and features of y = x^4.
Option 1 states: 'Local minimum: (10,0)  No inflection points.' In reality, y = x^4 reaches its lowest value at x = 0, where y = 0, so the local (and global) minimum is (0,0). The graph of x^4 is highly curved upwards and does not have any inflection points because its second derivative does not change sign across the domain. Therefore, this option misidentifies the minimum and correctly notes no inflection points, but the listed coordinate is incorrect.
Option 2 states: 'Local maximum:  Local minimum: (10, 0)  Inflection points: , (10, 0)'. First, a local maximum does not exist for y = x^4 since the function is increasing from the left of 0 to the right and has no peak; the curve opens upward like a cup. Second, the claimed local minimum at (10,0) is incorrect for the same reason as above: the actual minimum is at (0,0). Finally, there are no inflection points for x^4, so listing any inflection points is false. Overall, this option contains multiple incorrect claims.
Option 3 states: 'Local maximum:  Local minimum: (10, 0)  Inflection points:'. Again, there is no local maximum for x^4, and the minimum is not at (10,0) but at (0,0). The line ends without specifying any inflection points, which would be correct in terms of existence (there are none), but the minimum coordinate is wrong, making the entire option invalid.
Option 4 states: 'Local maximum:  Local minimum: (10, 0)  Inflection point:'. This option similarly asserts a non-existent local maximum and uses the incorrect minimum coordinate; furthermore, it mentions an inflection point in singular form, which would be inaccurate since there are no inflection points on y = x^4. The combination of a nonexistent maximum and wrong minimum coordinate confirms this option is incorrect.

Summarizing the reasoning across all options: each option either asserts a local maximum that cannot exist for x^4, or gives the wrong coordinates for the minimum, or misstates the presence of inflection points. Since the correct mathematical features for y = x^4 are a local (and global) minimum at (0,0) and no inflection points, none of the provided answer choices accurately reflects the actual graph features. The core misconception in each choice is the misplacement of the minimum at (10,0) and/or the presence of a maximum, which contradicts the geometry and calculus of x^4.

Sketch the graph and show all local extrema and inflection points. y = x4 单项选择题

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