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f(x)=(5x+1)(4x-8)(x+6) has zeros at 
x=-6,x=-(1)/(5), and 
x=2.
What is the sign of 
f on the interval 
-(1)/(5) < x < 2 ?
Choose 1 answer:
(A) 
f is always positive on the interval.
(B) 
f is always negative on the interval.
(C) 
f is sometimes positive and sometimes negative on the interval.

f(x)=(5x+1)(4x8)(x+6) f(x)=(5 x+1)(4 x-8)(x+6) has zeros at x=6,x=15 x=-6, x=-\frac{1}{5} , and x=2 x=2 .\newlineWhat is the sign of f f on the interval 15<x<2 -\frac{1}{5}<x<2 ?\newlineChoose 11 answer:\newline(A) f f is always positive on the interval.\newline(B) f f is always negative on the interval.\newline(C) f f is sometimes positive and sometimes negative on the interval.

Full solution

Q. f(x)=(5x+1)(4x8)(x+6) f(x)=(5 x+1)(4 x-8)(x+6) has zeros at x=6,x=15 x=-6, x=-\frac{1}{5} , and x=2 x=2 .\newlineWhat is the sign of f f on the interval 15<x<2 -\frac{1}{5}<x<2 ?\newlineChoose 11 answer:\newline(A) f f is always positive on the interval.\newline(B) f f is always negative on the interval.\newline(C) f f is sometimes positive and sometimes negative on the interval.
  1. Identify Zeros: Since f(x)f(x) has zeros at x=6x=-6, x=15x=-\frac{1}{5}, and x=2x=2, we can test the sign of f(x)f(x) by picking a test point from the interval 15<x<2-\frac{1}{5} < x < 2.
  2. Select Test Point: Let's pick x=0x=0 as our test point since it's easy to work with and it's within the interval.
  3. Evaluate f(x)f(x): Plug x=0x=0 into f(x)f(x) to see the sign: f(0)=(50+1)(408)(0+6)=(1)(8)(6)f(0)=(5\cdot0+1)(4\cdot0-8)(0+6) = (1)(-8)(6).
  4. Calculate Sign: Calculate the sign of f(0)f(0): (1)(8)(6)=48(1)(-8)(6) = -48, which is negative.
  5. Final Conclusion: Since f(0)f(0) is negative and there are no zeros between 15-\frac{1}{5} and 22, f(x)f(x) must be negative for all xx in the interval 15<x<2-\frac{1}{5} < x < 2.

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