sketch the graph of the polynomial function f(x)=-x^3-5x^2

Question

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Answer 1

f(x)=-x^2(x+5). The curve passes through the origin (0,0) which is a local maximum. It cuts the x axis at -5 and it has a minumum when f'(x)=0=-3x^2-10x=-x(3x+10), so at x=-10/3, f(x)=-100/9(-10/3+5)=-100/9*5/3=-500/27=-18.52 approx. So (-10/3,-18.52) is a minimum. When x is large and negative f(x) is large and positive, so the graph on the left descends to cut the x axis at -5 and continues to the minimum then rises to touch the origin before descending again. When x is large and positive f(x) is large and negative.