Let f′(x) = 100x(2x+3)(x−5).Use sign chart to determine the interval(s) on which f is increasing or decreasing?

f(x) is increasing/decreasing there where its derivative is + or - resp. 

So, let its derivative f '(x) = g(x).

You Are told that g(x) = 100x(2x+3)(x−5).

Where is this positive & negative? You should be able to answer that because such a question is early high school algebra and has nothing to do with Calculus.

So, Try to recall your high school algebra and tell us where 100x(2x+3)(x−5)

Is positive and negative.

Hopefully no one will spoil you the answer. That would be very irresponsible of them. And don't forget to vote me best answer for being the first to correctly walk you through without spoiling the answer. That way it gives you an honest chance to work at it and to get good at it. You are welcome!

I don't know what a sign chart is, but I can tell you the intervals on which f is increasing or decreasing.

The points where f'(x) = 0 are at x = 0, x = -3/2, and x = 5.

So the intervals to consider are (-infinity, -3/2), (-3/2, 0), (0,5), and (5,infinity).

When x > 5, all three factors in the factored expression for f'(x) are positive, so f is increasing on (5, infinity).

When 0 < x < 5, the factor x - 5 is negative but the other two factors are both positive, so f'(x) is negative and f is decreasing on (0,5).

When -3/2 < x < 0, the factor 2x + 3 is positive but the other two factors are both negative, so f'(x) is positive, and f is increasing on (-3/2,0).

When x < -3/2, all three factors are negative, so f is decreasing on (-infinity, -3/2).