Respuesta :
Answer:
a) The critical points are [tex]x = 3[/tex] and [tex]x = -6[/tex].
b) f is decreasing in the interval [tex](-\infty, -6)[/tex]
f is increasing in the intervals [tex](-6,3)[/tex] and [tex](3,\infty)[/tex].
c) Local minima: [tex]x = -6[/tex]
Local maxima: No local maxima
Step-by-step explanation:
(a) what are the critical points of f?
The critical points of f are those in which [tex]f^{\prime}(x) = 0[/tex]. So
[tex]f^{\prime}(x) = 0[/tex]
[tex](x-3)^{2}(x+6) = 0[/tex]
So, the critical points are [tex]x = 3[/tex] and [tex]x = -6[/tex].
(b) on what intervals is f increasing or decreasing? (if there is no interval put no interval)
For any interval, if [tex]f^{\prime}[/tex] is positive, f is increasing in the interval. If it is negative, f is decreasing in the interval.
Our critical points are [tex]x = 3[/tex] and [tex]x = -6[/tex]. So we have those following intervals:
[tex](-\infty, -6), (-6,3), (3, \infty)[/tex]
We select a point x in each interval, and calculate [tex]f^{\prime}(x)[/tex].
So
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[tex](-\infty, -6)[/tex]
[tex]f^{\prime}(-7) = (-7-3)^{2}(-7+6) = (100)(-1) = -100[/tex]
f is decreasing in the interval [tex](-\infty, -6)[/tex]
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[tex](-6,3)[/tex]
[tex]f^{\prime}(2) = (2-3)^{2}(2+6) = (1)(8) = 8[/tex]
f is increasing in the interval [tex](-6,3)[/tex].
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[tex](3, \infty)[/tex]
[tex]f^{\prime}(4) = (4-3)^{2}(4+6) = (1)(10) = 10[/tex]
f is increasing in the interval [tex](3,\infty)[/tex].
(c) At what points, if any, does f assume local maximum and minima values. ( if there is no local maxima put mo local maxima) if there is no local minima put no local minima
At a critical point x, if the function goes from decreasing to increasing, it is a local minima. And if the function goes from increasing to decreasing, it is a local maxima.
So, for each critical point is this problem:
At [tex]x = -6[/tex], f goes from decreasing to increasing.
So [tex]x = -6[/tex], f assume a local minima value
At [tex]x = 3[/tex], f goes from increasing to increasing. So, there it is not a local maxima nor a local minima. So, there is no local maxima for this function.
