The function f(x)=x/(x²-6 x-16), x ∈ ℝ-{-2,8}

Solution

Maths sir · 3 steps

Differentiate with the quotient rule and look at the numerator: it can never be positive.

1. Quotient rule. Write f(x) = x/(x²-6x-16). The numerator of f'(x) is (x²-6x-16)· 1 - x(2x-6), and the denominator is (x²-6x-16)².

   f'(x) = (x²-6x-16-2x²+6x)/(x²-6x-16)² = -(x²+16)/(x²-6x-16)²

2. Check the sign. x²+16 ≥ 16 > 0 for every real x, so the numerator is always negative. The denominator is a square, so it is positive for every x ≠ -2, 8. That makes f'(x) < 0 everywhere in the domain.
3. So. f is decreasing on each of (-∞,-2), (-2,8) and (8,∞) separately, which is option (a). It is decreasing on each piece, not across the breaks at x=-2 and x=8, where the graph jumps.

The idea to keep. For a rational function, find f' and factor its numerator. If that numerator has one fixed sign, the function is monotonic on every interval of its domain, one piece at a time.

Answer from the NTA final answer key. Solution by PrepWiser.