[amp_mcq option1=”local maximum” option2=”local minimum” option3=”both local maximum and minimum” option4=”neither local maximum nor local minimum” correct=”option1″]<\/p>\n
The function $f(x) = x^3$ has a local minimum at $x = 0$.<\/p>\n
A local minimum is a point in the graph of a function where the function value is less than or equal to the function values in the surrounding area. In other words, it is a point where the function is “going down” before it starts to “go up” again.<\/p>\n
The function $f(x) = x^3$ is increasing for all real numbers. This means that the function value is always increasing as $x$ increases. However, the function value is increasing at a decreasing rate. This means that the function is “going up” more slowly as $x$ increases.<\/p>\n
At $x = 0$, the function value is $0$. This is the lowest value that the function takes on. Therefore, $x = 0$ is a local minimum.<\/p>\n
The other options are incorrect because they do not accurately describe the behavior of the function at $x = 0$.<\/p>\n
Option A: local maximum. A local maximum is a point in the graph of a function where the function value is greater than or equal to the function values in the surrounding area. In other words, it is a point where the function is “going up” before it starts to “go down” again. However, the function $f(x) = x^3$ is increasing for all real numbers. This means that the function value is always increasing as $x$ increases. Therefore, there can be no local maximum at $x = 0$.<\/p>\n
Option B: local minimum. This is the correct answer.<\/p>\n
Option C: both local maximum and minimum. This is incorrect because the function $f(x) = x^3$ has only one local minimum, at $x = 0$.<\/p>\n
Option D: neither local maximum nor minimum. This is incorrect because the function $f(x) = x^3$ has a local minimum at $x = 0$.<\/p>\n","protected":false},"excerpt":{"rendered":"
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