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Given f '(x) = (5 - x)(8 - x), determine the intervals on which f(x) is increasing or decreasing. (2 points)

Decreasing (-∞, 5); increasing on (8, ∞)
Increasing (-∞, -5) U (-8, ∞); increasing on (-5, -8)
Decreasing (5, 8); increasing on (-∞, 5) U (8, ∞)
Decreasing (-∞, 5) U (8, ∞); increasing on (5, 8)

Is it B?

Respuesta :

LammettHash
[tex]f'(x)=0[/tex] when either [tex]x=5[/tex] or [tex]x=8[/tex]. The intervals you need to check are then [tex](-\infty,5)[/tex], [tex](5,8)[/tex], and [tex](8,\infty)[/tex].

This means you can ignore the option with intervals containing [tex]-5[/tex]; this number is irrelevant. So (B) would not be correct.

[tex]f(x)[/tex] will be increasing on the interval to the left or right of each critical value if the sign of the derivative in that interval is positive, or decreasing if the sign is negative.

For example, from the interval [tex](-\infty,5)[/tex] we could pick [tex]x=0[/tex]. Then [tex]f'(0)=5\times8=40>0[/tex], so [tex]f(x)[/tex] is increasing on this interval. Luckily this is all you need to check, because (C) is the only option involving this conclusion.

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