Question

Let f(x)=x^3-6x^2+3x+10f(x)=x 3
−6x 2
+3x+10, then choose the set of correct options regarding f(x)f(x).
If x \in [-2, 2] \cup (5, \infty)x∈[−2,2]∪(5,[infinity]), then f(x)f(x) is positive.
If x \in (-1, 4] \cup (5, \infty)x∈(−1,4]∪(5,[infinity]), then f(x)f(x) is positive.
If x \in [0, 1] \cup (5, \infty)x∈[0,1]∪(5,[infinity]), then f(x)f(x) is negative.
If x \in [0, 1] \cup (10, \infty)x∈[0,1]∪(10,[infinity]), then f(x)f(x) is positive.
If x \in (-\infty, -2] \cup (2, 5)x∈(−[infinity],−2]∪(2,5), then f(x)f(x) is negative.

Answer 1

Step-by-step explanation:

If x \in (-1, 4] \cup (5, \infty)x∈(−1,4]∪(5,∞), then f(x)f(x) is positive.

Answer 2

If x belongs to (-1, 4] U (5, infinityty)x belongs to (−1,4]U(5,infinity), then f(x)f(x) is positive.

Explanation:

• The sign of a polynomial between any two consecutive zeros is either always positive or always negative
• This is because polynomial functions are continuous functions (no breaks in the graph), which means that the only way to change signs is to cross the x-axis. But if this happened, the given zeros would not be consecutive
• It is not necessary, however, for a polynomial function to change signs between zeros.