make a conjecture about the behavior of the graph?

Question

make a conjecture about the behavior of the graph of
y = x^3(3x-4)(x+2)^2
in the vicinity of its x-intercepts

Answer 1

The x-intercepts occur at x =0, x =4/3, and x=-2. The root x =0 has multiplicity 3, which is odd, so at that point the graph should be tangent to the x-axis, cross the x-axis, and have an inflection point there. The root x = -2 has multiplicity 2, which is even, so the graph should be tangent to but not cross the x-axis there. The root x = 4/3 is simple, so at that point the curve should cross the x-axis without being tangent to it. All of this conclusion is consistent with the graph, should you use a graphing calculator or whathaveyou.

Answer 2

The roots (x-intercepts) are 0, 4/3 and -2.

At any root where the exponent is odd, the graph cuts through the x-axis. The root itself is generally an inflection point. In this function, this would happen at x = 0 and x = 4/3.

At any root where the exponent is even, there is a relative maximum or minimum value. This would happen at x = -2. You can tell whether this is a maximum or a minimum by comparing the y-value at -2 with the y-value at other points nearby (which you can test on a calculator). In this case, I believe it is a relative minimum.