Answer to Question #244770 in Calculus for N1yxz

Question

Let f(x)=x^3-6x^2+3x+10, then choose the set of correct options regarding f(x).

• If x element of (-\infinity, 1] union (2, 5) x∈(−∞,1]∪(2,5), then f(x) is negative.

• If x element of [-2, 2] union (5, \infinity)x∈[−2,2]∪(5,∞), then f(x) is positive.

• If x element of [0, 1] union (10, \infinity)x∈[0,1]∪(10,∞), then f(x) is positive.

• If x element of [2, 3] union (5, \infinity) x∈[2,3]∪(5,∞), then f(x) is positive.

• If x element of (-1, 2) union (5, \infinity) x∈(−1,2)∪(5,∞), then f(x) is positive.

Answer 1

$$f(x)=0=>x^3-6x^2+3x+10=0$$

$$x^2(x+1)-7x(x+1)+10(x+1)=0$$

$$(x+1)(x^2-7x+10)=0$$

$$(x+1)(x-2)(x-5)=0$$

$$x_1=-1, x_2=2, x_3=5$$

If $x<-1,$ then $f(x)<0.$

If $-1<x<2,$ then $f(x)>0.$

If $2<x<5,$ then $f(x)<0.$

If $x>5,$ then $f(x)>0.$

Answer:

5. If $x\in(-1, 2)\cup(5,\infin),$ then $f(x)$ is positive.