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f(x)=(x-5)(2x+7)(7x-3) has zeros at x=-3.5,x=(3)/(7), and x=5. What is the sign of f on the interval (3)/(7) < x < 5 ? 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)=(x−5)(2x+7)(7x−3) f(x)=(x-5)(2 x+7)(7 x-3) has zeros at x=−3.5,x=37 x=-3.5, x=\frac{3}{7} , and x=5 x=5 .\newlineWhat is the sign of f f on the interval \( \frac{3}{7}

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Q. f(x)=(x−5)(2x+7)(7x−3) f(x)=(x-5)(2 x+7)(7 x-3) has zeros at x=−3.5,x=37 x=-3.5, x=\frac{3}{7} , and x=5 x=5 .\newlineWhat is the sign of f f on the interval 37<x<5 \frac{3}{7}<x<5 ?\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 Function Zeros: Since f(x)f(x) has zeros at x=−3.5x=-3.5, x=37x=\frac{3}{7}, and x=5x=5, we know that the function changes sign at each of these points.
  2. Determine Interval Sign: To determine the sign of f(x)f(x) on the interval \frac{3}{7} < x < 5, we can pick a test point between 37\frac{3}{7} and 55. Let's pick x=4x=4.
  3. Select Test Point: Plug x=4x=4 into f(x)f(x) to see the sign: f(4)=(4−5)(2⋅4+7)(7⋅4−3)=(−1)(15)(25)=−375f(4)=(4-5)(2\cdot4+7)(7\cdot4-3) = (-1)(15)(25) = -375.
  4. Calculate Function Value: Since f(4)f(4) is negative, and there are no zeros between 37\frac{3}{7} and 55, f(x)f(x) is always negative on the interval \frac{3}{7} < x < 5.

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