Is #f(x)=(x-2)^2-6x-7 # increasing or decreasing at #x=-2 #?

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Answer 1 · 2016-01-28T20:52:05

Decreasing.

First, we should simplify #f(x)# by expanding #(x-2)^2#.

#f(x)=(x^2-4x+4)-6x-7#

#f(x)=x^2-10x-3#

To determine if the function is increasing or decreasing at #x=-2#, find the sign of the derivative at #x=-2#.

To find #f'(x)#, use the power rule on #f(x)#.

#f'(x)=2x-10#

Determine the sign of the derivative:

#f'(-2)=-2(2)-10=-14#

Since this is #>0#, the function is decreasing at #x=-2#.

We can check a graph:

graph{x^2-10x-3 [-5, 15, -35.9, 50]}