Assume that f is continuous on [-4,4] and differentiable on (-4,4). The table gives some values of f'(x) x: -4, -3, -2, -1, 0, 1, 2, 3, 4 f'(x): 74, 39, 10, -6, -14, -12, 0, 22, 55 a) estimate where f is increasing, decreasing, and has local extrema.

Respuesta :

[tex]f[/tex] will be increasing on the intervals where [tex]f'(x)>0[/tex] and decreasing wherever [tex]f'(x)<0[/tex]. Local extrema occur when [tex]f'(x)=0[/tex] and the sign of [tex]f'(x)[/tex] changes to either side of that point.

[tex]f'(x)[/tex] is positive when [tex]x[/tex] is between -4 and some number between -2 and -1, and also 2 (exclusive) and 4, so you can estimate that [tex]f(x)[/tex] is increasing on the intervals [-4, -2] and (2, 4].

[tex]f'(x)[/tex] is negative when [tex]x[/tex] is between some number between -2 and -1, up to some number less than 2. So [tex]f(x)[/tex] is decreasing on the interval [-1, 1].

You then have two possible cases for extrema occurring. The sign of [tex]f'(x)[/tex] changes for some [tex]x[/tex] between -2 and -1, and again to either side of [tex]x=2[/tex].
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