Question

Describe the graph of the function at its roots. f(x) = (x − 2)3(x + 6)2(x + 12) At x = 2, the graph crossesdoes not intersecttouches the x–axis. At x = −6, the graph crossesdoes not intersecttouches the x–axis. At x = −12, the graph crossesdoes not intersecttouches the x–axis.

Answer 1

Answer:the first option is crosses then touches lastly crosses again

Step-by-step explanation:

Answer 2

This type of problem can be decided by looking at whether the function is positive or negative in the vicinity of the zeros. There are three roots: -12, -6, and 2 at which the function is clearly 0. Look at the four intervals the x axis:

interval x<-12: function is negative (just test using a number <-12, say, -13)

interval (-12,-6): function is positive

interval (-6, 2): function is negative

interval x>2: positive

From the above you deduce that at the roots the function must be crossing the x-axis (as opposed to just touching it) because the function value changes its sign every time.