What does the graph of g(x) look like given that #f(x)=(d/(dx))g(x)# and #g(0)=0#?

Given f(x) as shown, sketch a graph of g(x) such that #f(x)=(d/(dx))g(x)# and #g(0)=0#
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1 Answer
Noah G
May 29, 2018

Since #g'(x) = f(x)#, #int_0^x f(x) dx = g(x)#.

We should be looking for points when #f(x) = 0# as these will be maximums and minimums. Furthermore, concavity will be determined by whether #f# is increasing or decreasing.

#f(x) = 0# twice. On the left one, the point will be a minimum since #f(x)# goes from negative (decreasing) to positive (increasing). The right one will be a maximum since it goes from positive (increasing) to negative (decreasing).

There will be a point of inflection from concave up to concave down where #f# flattens. In the end you should get a graph resembling the following.

enter image source here

Hopefully this helps!

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