Question

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

>If x∈(−∞,−1)∪(3,4), then f(x) is negative.

>If x∈(−1,4]∪(5,∞), then f(x) is positive.
>if x∈[2,3]∪(5,∞), then f(x) is positive.

>If x∈[0,1]∪(5,∞), then f(x) is negative.

>If x∈[0,1]∪(10,∞), then f(x) is positive​

Answer 1

Answer:

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Answer 2

Given: f(x)=x^3-6x^2+3x+10

To find: Range of x for which f(x) is negative or positive.

Solution: Here, we are given a cubic polynomial, and any such polynomial will have in general 3 zeroes. These can be the cases for the zeroes;

• All three zeroes might be real and distinct
• All three zeroes might be real, and two of them might be equal.
• All three zeroes might be real and equal.
• One zero might be real and the other two non-real (complex).

Now we need to find factors of the given polynomial.

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

f(x) = 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

x1= -1, x2= 2, x3= 5

x<-1 ,f(x) <0

-1<x<2, f(x) >0

2<x<5, f(x) <0

x>5, f(x) >0

So ,we can say that for if x ∈ (-1, 2) U(5, ∞) then f(x) is positive.

And if x ∈ ( -∞, -1) U (2,5)then f(x) is negative.

Therefore, A) If x∈(−∞,−1)∪(3,4), then f(x) is negative.