How do you find the intervals of increasing and decreasing using the first derivative given #y=-(x^2+8x+12)#?

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Answer 1 · 2017-02-06T08:04:47

The interval of increasing is #x in ]-oo, -4]#
The interval of decreasing is #x in [-4, +oo[#

We calculate the first derivative and then build a sign chart

Let #f(x)=-x^2-8x-12#

#f'(x)=-2x-8#

Critical point when #f'(x)=0#

#-2x-8=0#, #=>#, #x=-4#

Let construct the sign chart

#color(white)(aaaa)##x##color(white)(aaaaaaa)##-oo##color(white)(aaaa)##-4##color(white)(aaaaa)##+oo#

#color(white)(aaaa)##-2x-8##color(white)(aaaaaa)##+##color(white)(aa)##0##color(white)(aa)##-#

#color(white)(aaaa)##f'(x)##color(white)(aaaaaaaaa)##+##color(white)(aa)##0##color(white)(aa)##-#

#color(white)(aaaa)##f(x)##color(white)(aaaaaaaaaa)##↗##color(white)(a)##0##color(white)(aa)##↘#

The interval of increasing is #x in ]-oo, -4]#

The interval of decreasing is #x in [-4, +oo[#